You might want to look into a little-known MMO called 'World of Warcraft'. Of the 7 Healer specs available to play, 3 are explicitly designed with 'do damage as part of your healing gameplay' in mind (Disc Priest, Mistweaver, Holy Paladin), and the other four have talents/abilities that massively feed into allowing you to do damage on the side while your HOTs are ticking/people don't need healing (Resto Druid can Catweave and have up to like 6 DOTs active at once potentially. Holy Priest has Empyreal Blaze to blast out heavy hitting Holy Fires and its new Hero Talent tree, Archon, allows it to use their not-Assize, Halo, a lot more often which deals damage that adds up. Resto Shaman has a talent that makes their Healing Rain (Asylum but a GCD and can have 100% uptime) also deal damage over time to enemies in it, and has totems that can deal damage passively akin to old SMN Egi Autoattacks).
The only healer that is remotely close in DPS design to FFXIV healers is Preservation Evoker, with a nuke (Living Flame, which also functions as their Cure1), a 30s CD (Flame Breath), a gauge spender that competes for healing resources (Disintegrate) and a 2min burst damage attack (Deep Breath). But even then, Flame Breath can be augmented with a talent that makes it take 80% of the damage it deals, and apply that as healing to an injured ally. Or a talent that makes it so that when you hit enemies with Flame Breath, your Living Flame hits X more targets on the next use, with X being equal to the enemy count hit by Flame Breath (up to 4 bonus targets iirc)
The idea that 'healers heal and that's their whole thing' is a cancer that should have stayed dead after Cataclysm's healing requirements were received so... divisively. M6S has conclusively shown that this playerbase simply isn't prepared for the idea of 'healers heal', and the funniest part is how nobody who says 'healers should heal' never seems to realize that, once we have gear, the amount of healing we need to do will sharply decrease, and we'll be back where we started.
Surely it'd be more sensible to have a small DPS rotation for the Healers, comprised of several actions of such similar potency that A: there is an 'optimal rotation' for the sweats, but B: the difference between 'optimal' and 'smash Broil over and over and ignore everything else' is so small that it doesn't ever cause an Enrage? Furthermore, having more damage buttons for the inevitable downtime would allow more variety in how those DPS buttons interlink with our healing output, allowing for more design space for the Devs to work with. They said pre-Endwalker 'we didn't really know what to do with SCH' when showing off Expedient. Maybe if they hadn't removed all the damage buttons and gone scorched earth on the idea of Healers pressing non-Heal buttons, they'd have ideas to work with?
Here is an example of what I came up with for WHM, while trying to keep the number of new buttons added to the Job low (to lower dev-time), keep the core of the job as intact as possible (to lower dev-time/retain familiarity), but still improve the DPS rotation in downtime. With just two buttons, Banish (15s CD) and Blessing of the Elementals (Costs 50 of a new gauge, built by casting literally any MP-cost spell), and reducing Dia from 30s down to 12s, this design's 'optimal damage rotation' per 2mins goes from having 35 Glare3's, all the way down to just 17. I even added some diagrams to illustrate this difference, and the best part is, with that design, over 60% of the GCDs a WHM would use for their damage rotation would be instantcasts. If BLM's justification was 'it needed to have the changes so it could keep up with the mobility demands of the new fight design', then this WHM design would fit perfectly with that thinking. It also requires zero actual hotbar spaces to be added, because those two required buttons (Banish/BOTE) could be gained by simply merging Cure1 with Cure2, and Medica1 with Medica2, given that we never use Cure1 after we get Cure2 anyway, and Medica1 is equal in potency to 'Medica2's cast, plus a single tick'. So, just frontload one tick of Medica2, reduce the duration by 3s, and boom, hotbar slot saved. It's surprisingly easy to come up with ideas on how to use the current designs as a foundation, and build upon them, when you don't arbitrarily handicap yourself with restrictions like 'Healers should not be allowed a DPS rotation (but Tanks ARE allowed one, despite being equally 'not-a-DPS'???)
Also, if people really want to follow the flawed mantra of 'healers should only heal', they might be interested in my SGE design where Diagnosis and Prognosis have no MP cost. Spam overheal to your heart's content!