I think a lot of players here are over-romanticising the place that Healers tend to have in other MMO-and/or-RPGs.
Healers being minimised and pushed aside by high-skill, optimisation-oriented players is not new, it's not unique to FFXIV, and it's not specifically a consequence of FFXIV's design.
The Healing role, by its fundamental nature, has a target painted on its forehead "from the word go" — it's a role whose existence is based solely on facilitating the continued function of other roles.
To do this, Healing trades out the ability to do "other stuff" for the ability to prolong everyone's time "in the ring" before getting knocked-out.
That means, inherently, that Healers are basically just "Your role, but worse", with the loss ostensibly filled-in by Healing tools.
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But while other roles have "infinite scaling" — ie, doing more DPS is always desirable and, conceptually, "possible" — Healing hits a flat wall: at some point, no one is going to KO, and then you don't need more healing, because more healing is useless if everyone's risk of KO has already been alleviated.
To digress a second: This is not easy to solve.
For example, say that you allow "overhealing" — kind of like Repairing your gear up to 200%, you can heal people "past" 100% HP, and this becomes more and more "doable" with better gear (or, etc).
...That's cute in theory, but what's the purpose? You either need that to survive — in which case it's basically just an extension of the normal healing/mitigation process — or you don't need it to survive, in which case it will immediately get cut and ignored in favor of more DPS.
Or let's say that healing beyond 100% causes the overheal amount to be dealt as damage to a nearby target.
That's also cute in theory, but it's not actually making "healing" more useful — it's just an alternative DPS rotation.
And so on...
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And that means that any time that a combination of utility tools can accomplish that same goal, Healers are perceived as having nothing left to offer to justify a slot, other than weaker versions of what someone else could do — so, they get replaced.
This is not unique to FFXIV; I actually can't think of a similar game, off the top of my head, where high-skill / optimising players won't try to squeeze out as many Healers as possible from their comps, and sometimes succeed.
Fundamentally, while Healer-oriented players like for their Role to feel clever, desired, and heroic, other players usually view more like a parent: very useful during the period where it's needed, and then immediately marginalised as much as possible (in terms of presence and influence) by its former "wards" once it's... not needed any more.
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ie: Healers (usually?) love to feel needed — but other Roles love to not need Healers.
This is a fundamental tension, and one that relies on basically forcefully-engineering a place for Healers into encounters, in direct opposition to the psychology of other players.
Basically, a "composition tax", so to speak: "You can complete this encounter, but in order to do so, you'll need to bring along Healer-type players, and allow them to 'save you' and have fun, too".
Usually this "bribery" is accomplished by making people die, since that's a powerful motivator.
But the cutting-edge type players will still constantly be calculating and plotting how they could "not die" in a more efficient manner, and thereby cut down on how much they're "taxed" in terms of composition — because Healers just fundamentally "bring less" to a party when you ignore intentionally-engineered time-to-die mechanics.
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All of this is why the Healing design in XIV is such a huge mess: "Yoshida" (as a nebulous concept representing the design team as a whole) wants to allow "pure healing" to be a style, and to be what characterises the Healer Jobs.
And to "support" that, "he" insists on vomiting like Cuchulainn more and more redundant and superfluous healing-tools onto the Healer Jobs, from SB to present.
The problem is that this is done thoughtlessly, and so crashes directly into what I outlined above: If you just keep mindlessly pouring more and more Healing tools onto a Healer, without proportionally increasing the demand-pressure for healing, then all you do is accelerate the rate at which Healer as a role "overstays its welcome" in the calculating minds of top-end player.
ie — because it's easier and easier to meet healing demands, the justification for X number of Healers being there becomes more and more dubious.
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Of course, there's always also the temptation to try to give "cool toys" to non-Healer classes, in order to try to give them ways to "cleverly support" the team — since merely adding "Damage Button #27" can become conspicuously-stale.
But... you see the problem: If healers are already overloaded with more tools than they even need, then... adding additional Healing-style support tools to other Roles is only going to further accelerate the "decay rate" of Healer relevancy in a given piece of content...
...because giving such tools to non-Healer Roles ends up providing a more granular way for a cutting-edge team to fine-tune the balance between Healing presence and "other benefits" (eg, Damage, Mitigation, etc).
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And of course, other games attempt to solve this tension by simply engineering encounters to pour out overwhelming amounts of damage, making it excessively-difficult and/or numerically-impossible to remove Healers and replace them with the more minor, limited, or tone-down support tools available to other Roles as utility flavor.
But... the problem is, that's a razor's-edge to balance, because...
...Well, think of it in terms of DPS for a moment: Usually, it's "okay" to bring your friend who's playing at maybe 60-70% of their class's DPS potential, because encounter killtimes are usually "slushy".
However, if an enrage timer is suddenly tuned mercilessly-tight, then — after too many vexing 1-2% wipes — your 60%-performing friend is probably going to get a Zoom call from Vishal Garg soon.
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Healing faces the same issue: If you tune it tightly enough to allow the best-of-the-best Healers to shine decisively, and to force all Healers to fully-utilise every tool and cast-GCD to its limit, then... a lot of Healers in that 1-60% (or 70, or 80, or 90, etc) range just won't be able to keep up.
And because the "enrage" for healing is "not being able to keep up with damage intake"... if you tune healing tightly, then when you bring your 60%-performing Healer friend, other players just KO, causing frequent or prolonged interruptions to those players's ability to control and play their character at all, through (ostensibly) no fault of their own.
And that tends to quickly breed intense frustration, hostility, stress, resentment, etc.
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So, even in games with more demanding incoming damage philosophies, Healing still ends up being relatively "slushy" across most tiers of content — in terms of not being tuned even remotely-close to requiring truly, algorithmically-optimal Healing play in order to "succeed".
And thus, space is opened up once again for pushing-out Healers: either by bringing in the most cutting-edge Healers, who can maximise casts and tools (and thus potentially pull the duty of an entire other Healer, or etc), or by just exploiting the utility tools that inevitably get peppered across other Roles... same as occurs in XIV.
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So, in order to not create a severe exclusionary problem and a lot of "social PvP", Healing inevitably needs to be given larger margins of error, and some amount of technical excess in its toolkit vs. the damage intake that "most" (if not all) content produces.
Logically, then, it makes sense to offer Healers the same thing as other Roles, but in reverse — ie, if "other Roles" are allowed to dabble in Healing utility, then Healers are allowed to dabble in DPS utility.
This allows more skilled Healers to still have a goal to strive for and refinements to achieve, even if they've overcome the healing demands for a given piece of content.
...And, indeed, in other games, Healers tend to have a reasonable kit of damage options tacked-on — not as sophisticated as a "real" DPS, perhaps, but enough to give them at least a minor challenge to work on refining, and some entertainment even when Healing is covered adequately.
And this solution "scales well", because Healers who are struggling more can just focus more on Healing, rather than simply wiping their team — the differentiation between Healer skill levels, and the sense of "growth" as a Healer, is filled-in more by working-in the successful utilisation of other tools than by raw Healing capability.
(Setting aside that resource-management — the tension between pouring out high-power healing, and running out of fuel to do it — is also usually a significant part of Healer gameplay and refinement as well... something else that FFXIV has become terminally-allergic to)
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...And then we have FFXIV, which has aggressively gone out of its way to remove all other tools from Healers — right down to even having a DPS rotation to strive for.
This creates a problem: Either you reward Healers with equal DPS to everyone else, despite pressing ~1 button to perform that output... or you "tune down" Healers proportional to the simplicity of their damage "rotation".
The former option is strange, and would (to go to a hyperbolic extreme for example's sake) paradoxically-encourage replacing real DPS with Healers, since they can accomplish the same thing for less effort and distraction.
The latter option, however, only more heavily encourages swapping-out Healers ASAP once they're not "needed" any more, because they physically can't "pull their weight" any harder on DPS without creating a tuning nightmare by packing ridiculous power into their single damage button.
In actual practice, the XIV team seems to be trying to dodge both issues by just hard-coding mechanics to target the Healing role.
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So, I think the perspective should be kept a little more clear regarding this:
Trying to squeeze out dedicated Healer roles whenever the opportunity strikes is not unique to FFXIV, it's a fundamental problem that's inherent to the "RPG Healing" paradigm / design.
FFXIV's mangled design path, however, has perhaps served to exacerbate, magnify, and encourage this mentality to an increasing degree, as the Healer role continues to stagnate creatively, developmentally, and mechanically.
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As for potential solutions...?
Honestly, I don't see very many, assuming nothing significant changes about the overall design framework.
You can pump up damage intake. But this has limitations, because it's inherently exclusionary.
Now, you can certainly say, "Good, I want lower-skilled players to be excluded until they either get better, or give up and quit" — that's a philosophy that has definitely been prevalent throughout the existence of gaming (frankly, even long before modern, electronic gaming).
However, it's also clearly anathema to the current SEBU3shida philosophy, so probably a non-starter to begin with — and even if done, it will probably hit a hard wall sometime around Extremes, and not be extended any further "down" into content, leaving the same old mess in the rest of the game.
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Or you could dramatically pump up Healer rotational involvement, and DPS output in turn — bringing them much closer to, say, Dancer, and treating them more like "DPS-with-Utility" (ie, the current PvP approach), and thus reducing the temptation to cut them out of content when optimising.
This would fit very cleanly into the FFXIV combat system (such as it is).
However, this is also clearly anathema to the current development philosophies, because "Healers should be healers" (or... something), and it's apparently too confusing / threatening / traumatising / upsetting / ??? to White Mages in Japan if they have more than 1 damage key.
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Or, perhaps you could reposition Healing to more of a broad Support role — so that even if you don't need raw Healing, there's still a "good reason" to bring Healers.
For example, make Bravery a Healer Role Action — blah blah "partywide 10% damage increase, 60s cooldown" — and then also give each Healer its own version of something like both Chain Strategem and Cards, so that you're still incentivised to bring 2 Healers, if the game needs that kind of tidiness.
For example, White Mage Dia increases the target's damage taken by 5% while active, and White Mage gets a third kind of Lily (OGCD, and separate from the Afflatus Lilies) that increases the target's Critical Strike chance by 10% for 30s.
...But I can already see SE's consternation: "If we give Healers DPS-increasing support tools, then the Support DPS like Dancer will surely feel confused about their identity. We need to keep the Support DPS feeling meaningful, so we cannot do this. Please understand, thank you. Consider pressing Glare in Ultimates if you are seeking a challenge as Healer."
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...So, honestly, unless there's some fundamental shake-up inside FFXIV's design team or design philosophy, I kind of think Healers are just doomed, design-wise.