Cure III actually gets use in high end content pushing, single healer runs, and organize groups. It's considered one of the most powerful healing tools in the game.
to imply that we use them all equally at level 90
One doesn't imply something when they explicitly say it's not true. I noted that there's no perfect parity and that many are not used. But that's also true of WAR. For WAR's single target rotation, they don't use any of their AOE buttons. If WAR is Off-Tanking, they don't use most of their mitigation buttons. An OT WAR will use 1-2-3/4, Infuriate, Inner Release, Fell Cleave, Primal Rend, Upheaval, and probably Nascent Flash and Shake It Off/Reprisal (rarely). WHM will use Glare and Dia, of course, but also Solace, Rapture, Misery, Assize, Asylum, Temperance, Divine Benison, Tetra, Bene, Lilybell, Presence of Mind, Aquaveil, Plenary Indulgence, Swiftcast, Lucid, and Thin Air, and occasionally Medica 2 or Regen (for bleed heavy fights)
...that's a WHM not using any of their hardcast GCD heals. That's 20 buttons for WHM vs 12 for WAR. WAR has several buttons change to others based on context, and more of their buttons are GCDs. That's the difference.
If things like Assize, Asylum, Temperance, and so on were GCDs, then we'd have a closer parity. There'd be less Glare spam because you'd have to burn those GCDs on actual healing and defensive actions.
...and people would hate it. Thus we get Glarespam instead.
healers are really some of the only jobs (along with BLM, SMN, and MNK) which have redundant abilities on their hotbars.
Shield Bash, arguably Clemency (in high end raider forums), most AOE buttons (not used in Extremes/Savages)...I dunno, probably more than just Healers.
SE have no intentions of increasing healing requirements,
They literally just did this in the most recent raid tier. I don't think "have no intentions" is a statement of fact when it's been so recently disproven.
Going FORWARD, they may not do it again because of how much the community complained about it, but that's on the community. The community would also complain if we made the healer DPS kits more complex, so that's not a good counter argument.
That said:
I do agree that Cure should upgrade to Cure 2 (with a MP cost of 500 - Cure 2's MP cost is silly considering WHM is supposed to be the powerful throughput healer of the bunch), and I've proposed myself Aero 3 be added to the game as something other than Holy you can press in dungeons. Something that's a damage gain on 2/3+ targets like Holy is.
Medica has a use and gets a lot of use in 24 mans. Seriously, was doing one last night and so many people were eating vuln stacks I had no resources at any given time, and once you cast a Medica 2 and people are still standing in bad, your options are spot heal 5 party members with Cure 2, cast Medica, or stupidly cast Medica 2 over and over again while the HoT is still ticking. Medica is the answer to that.
Now, if Medica 2 (MP cost 900 instead of 1000, Potency 400 instead of 250 with a 150 Regen) where chain casting it was equally as effective as chain casting Medica 1, then a straight upgrade would make sense.
Deleting heals on a healer so you can make room for damage spells on a healer just seems weird.
Besides, we can go the WAR approach and just have buttons change to other buttons. Like after casting Dia, your next Glare could be "High Glare" that does 200% damage and has a more blinding animation. Likewise, Aero 3 could make the next Holy "High Holy" that uses the RDM LB3 animation.