As things stand, the healers are made to basically be DPS turrets that have a few special mechanics tailored to them, and to fix other people's mistakes. Both of those latters being recovering HP, barring the occasional damage prevention.
Fundamentally, I believe that how the weights of these are placed is the problem in FFXIV. As things stand, the priority of how duties is made is dealing damage, fixing mistakes, and doing healer mechanics. Yet at the same time, the job design prioritizes fixing mistakes, healer mechanics, then dealing damage. The complete reverse.
The balance might change from duty to duty, but to my observations at least, it feels like this is how the encounters and jobs are designed, especially in ShB.
Now, as a healer main, I do, in fact, enjoy the idea of being the one keeping everyone alive. I don't know how many people do it for the same reason, but I enjoy the power fantasy of being the very reason why everyone else is still alive after every fight. If I don't heal, the tank dies. If I don't heal, the DPS dies. Everything is enabled by the fact that I heal. Each role has their own fantasy, but mine is that I chose who continues to exist and who doesn't.
So due to that, I'm not trying to advocate for making healers more like DPS or something, but to fix some underlying design directions when it comes to healing.
Now, this disconnect between encounter design and role design has been quite an issue lately, though it is one that had been slowly growing for a long while, and personally, I think that it stems from the meaning of the healer role.
What I mean is that you can view healers in two basic ways: the first is as a tool to fix the errors of other players. Someone gets hit by an avoidable mechanic? Heal them before they get hit again. This is the most wide spread view of healers, and the cause of the old meme: healers adjust.
The latter way to view healers is as a method of dealing with enemy aggression, similar to how tanks work. The enemy deals unavoidable damage, it's the healer's job to make sure that everyone's in a position to survive the next one.
Personally of the two, the latter one is an extremely healthy view of healers. Rather than a tool to fix others mistakes, prioritizing on being the tool that enables others to continue doing their jobs is a much more interesting and friendly way for healers to exist.
By prioritizing on the latter view, everyone becomes more aware of their own role, and has to take responsibility of themselves. Though there has to be a few changes to encounters to facilitate this.