There's 2 major issues; healers end up pressing a singular button over and over, and that their presence could largely be removed with little loss outside people standing in the orange.
You could fix the latter problem by nerfing or removing non-healer heals, although that brings in the ire of those who main those jobs for removing a tool they may enjoy, not an ideal fix. You could instead increase the amount of unavoidable damage that any piece of content does, but now you're at the mercy of people making sure they actually use those abilities rather than dumping all that responsibility on the healer, who may not be able to keep up with that extra damage that may be tuned for those extra heals from non-healers.
The fix for this isn't as simple as choosing one over the other, you'll have to take small doses of both; slight nerfs to non-healer heals (primarily WAR, followed by PLD), a slight increase in incoming damage (unavoidable damage in particular), and personally, a slight hit to tank defence (that's a discussion for another time).
For fixing the constant nuke presses, every oGCD used is a GCD heal not used, that GCD then gets spent on the primary nuke in its place, so it stands to reason that the less oGCDs we have access to, the more valuable our GCDs end up becoming, and the less we press the nuke. This has the problem of course of removing the heal potential that some older mechanics may have been tuned for, so we can't just remove oGCDs outright, but we could potentially nerf their availability with longer cooldowns or stricter resources.
Another fix for the constant pressing of the singular nuke is the ever popular DPS rotation; every healer gets a handful of unique attack spells that defines a playstyle rather than having them all copy-paste the nuke + DoT combo. This has its own issues in that some healers have very bloated kits as it is and may not have the room for more than 1 or 2 extra attack spells.
Like with the healing fixes, this likely needs a combination of both fixes; curb a handful of oGCD heals to make room for damage spells while also making the GCD heals more valuable in the process. This is a harder thing to fix as there'd likely need to be tests with older content to make sure those old heal checks can still be surpassed with any major changes to the healing kit.
"What healers truly want" is ultimately going to be very difficult because everyone has individual ideas and preferences on what they want; some are happy to just have an extra damage button or two, some just want more incoming damage to heal, while others want everything in between. You can't really "narrow down" what potentially thousands of players want; but you can pinpoint what they don't want, and we can at least point to the current healers and say "not that".
All this to say; yeah it's not simple fixes and people have different priorities in what they like about healers. You can blame Square for this after now 3 expansions of poorly designed healers.