I agree, I love a shitshow roulette. My favorite dungeon is probably Pharos Sirius because everyone kills the Zu eggs in the second boss which causes her to enrage and start body checking people for nearly their full HP and stunning them. On the topic of light or heavy damage, it really just comes down to semantics. Whether or not we're describing the damage as high or low, the point is that each healer has a surplus of healing power in contrast to the general amount of damage we take, even if that damage is taking the party down to around 40% HP. And to a point, this isn't a bad thing. It makes storyline content, dungeons, alliance raids, and normal raids, very accessible to novice and intermediate healers, as well as other light content like treasure map dungeons and Bozja. But it also means it's very easy for expert and master level healers can accomplish healing requirements with a fraction of their healing kit.
Which leads me to the point of viable healing tools. Because you're right, you do want more healing tools to feel viable all the time. Healing on an expert or master level renders the majority of our healing tools as "unviable" or "unnecessary" currently, which amounts to the majority of actions each healer can access. On tanks and DPS jobs, you have buttons like Bloodbath that aren't always necessary, but that's not a bad thing either, because most of DRG's kit it always viable and always necessary, you just also have a pocket full of situational tools that can help you in a pinch. Meanwhile, it feels like the opposite for healers. The majority of your hotbar actions are situational, and only a handful are consistently viable: your DPS tools, your buffs (if you have them), and your primary healing buttons, i.e. Kardia, Kerachole, and Ixochole on SGE, or Afflatus Rapture, Afflatus Solace, and Asylum for WHM (I may be slightly off on that one, but those are the main go-tos I see of WHM when I play DNC).
This is why the main argument I and several others continue to bring up is having things that are always useful regardless of whether or not healing is needed, because it should feel more like the tank and DPS jobs where you're engaging with most of your tools consistently with a handful of utility or niche tools. So when it comes to whether or not all GCD heals should be DPS neutral, I still don't think it should be that free. You should have a selection of tools that make your GCD healing viable and necessary for when damage is taken, but not every tool should be free like that. Like the example with Cure III that you mentioned... it's something that we could improve on, absolutely, but having some really powerful safety net heals that aren't optimal but are exceptional at getting the player out of a stick scenario, they can still have a lot of value even if they're not regularly engaged with. Something you could do to better balance this with WHM's GCD library could look something like this:
Cure 1 - Make it instant cast, and attach Esuna to it. It's niche, but it merges two already very infrequently needed tools into one action and makes it easier to use when needed.
Cure II - Make it instant cast, and have the potency scale with the target's missing HP. It's not DPS neutral, but a much more effective tool at rescuing a single party member.
Medica - Replace it with an upgraded new spell that's instant cast, restores HP in an AoE with light potency, and also deals neutral damage to nearby enemies, like a GCD Assize with a high MP cost. Lily heals are more efficient for viable, DPS neutral healing tools, especially because Misery can be used to gain DPS, but this makes it a potentially viable tool for when lilies have run out, and gives you another tool to use with Thin Air.
Cure III - Make it instant cast with an even larger radius, and also scales with the target's missing HP. Still a DPS loss, but an even better emergency tool.
Regen and Medica II - Replace both of these with one OGCD ability that makes your next GCD heal apply a regen that's stronger on single target spells. It makes it so that you can add regen effects to your afflatus heals, or this upgraded Medica button. You could even do something fun like make Aero/Dia tics have a chance to shorten the cooldown by a few seconds, making your DoT feel more connected to your gameplay rather than just being a DoT for DoT's sake.
- Afflatus Solace and Rapture - Give them a cast time. This makes the instant cast nature of your Cure spells more valuable in certain scenarios. Your optimal heals are no longer the easiest ones to use, but not hard to use either.
Something like that gives each heal a more clearly defined purpose, and makes clutch healing with Cure II and Cure III overall more worth the loss in uptime and especially useful for prog scenarios.