FFXIV by design will struggle to replicate that for a couple of reasons though.
In 'healer centric' MMOs (And even other RPGS such as Baldur's Gate TBH) like WoW and Everquest, you are always scaling yourself to the content. You're not forced to bring X amount of healers, instead you bring what you need. First time I killed Zlandicar in Everquest for Sleeper's keys we had a literal army of dorf clerics. When I killed him years later with some casual friends, we didn't even have a full group, I as the debuffing Shaman was also the sole healer and as such, I had plenty to do.
The other issue of course is that WoW and Everquest both worked on an attrition focused model. You didn't wipe to a 1 shot, you wiped because you ran out of resources or even just bodies.
IMHO the true solution needs to be tackled from multiple angles at once.
Just switching to attrition isn't great by itself as our healing kits aren't that much more satisfying than our damage kits, they are really just a vomit box of cooldowns that heal and little else. Even 3.4 AST with it's time extension mini games gives the current healing kits a run for their money IMO.
Chopping our oGCD heals into GCDs amplifies that issue making significant portions of the kit completely redundant without significant reworks. Why would a somewhat casual WHM press any of it when they could just press Medica II?
As I mathed out earlier, upping incoming damage alone isn't the answer either. With how things work in this game there's a very fine line where healing goes from being trivial to one shotting randoms and there's too little in between to get some satisfaction for healers outside of this tiny band.
If you ask me the solution is as follows:
Switch to an attrition model, at least double our HP, maybe even triple.
Revamp our kits top to bottom with a focus on ability consolidation, interaction within abilities in our kit and emphasising the importance of cooldown usage vs just pressing Medica II be it by resource limitations, accumulative buffs from cooldown usage or just raw throughput limitations. We have 4 healer jobs, try something different on each of them and see what sticks.
Come down hard on the availability of combat ressing. Forcing people to play a little safer is a good bump to lessening the value of raw damage above all else and will be a step in helping ease off the current need to set a fights difficulty by its enrage timer.
Obviously this isn't without it's problems. It's basically going to render old content almost entirely obsolete but frankly, I think it's something that needs to be done for the long term health of the game. The healing situation in FFXIV has progressively gotten worse and worse with every expansion post HW. Either they course correct and deal with the pain or they accept that healing in this game is forever condemned to be unsatisfying and unenjoyable outside of the narrow little window that is early Savage prog.
*Edit* Also, your Everquest comparison claiming that healers just sat isn't particularly fair. As a Cleric in a serious raid? Sure. But you could almost guarantee that said cleric in said serious raid was multi boxing another character. And as a Shaman main, I had enough to do that I generally couldn't multi box and I spent very little time sat down. Depending on the situation I could be baby sitting a mod rod team, pulling bodies, handling debuffs and potentially even CC or even fetching and queuing up mobs.