Ya know. I was thinking the same, and it never bugged me unless I played WHM, then it hit me.
That's what it all is. Sch has the fairy which lets them sit and DPS a bit more, not to mention that they carry some nice utility with chain, the ability to weave with Miasma 2, so on and so forth
AST has basic DPS, but lets the other DPS go nuts with their cards, contributing by playing Yu-Gi-Oh, and it's not like their DPS is bad.

I think that could be something for WHM to focus on. OGCD big heals to give to people while they sit there with active DPS. Not really asking for a rotation per se, though I do think it could work, but more so just more than stone. Maybe something proc based, maybe more OGCD attacks- use that water element and stuff? But yeah, more things like assize, and tetra. Forgo the "fat heals" healer because that just doesn't matter and is never good to have in this game, and just adopt the healer that doesn't have to hard-heal near as much. Adopt more OGCD means to heal via buffs and unique abilities. Maybe every three stones or so many ticks of aero your next cure is instacast? Maybe a cooldown to let you cast two skills at once at multiple targets? More things like Assize that heal and do damage?

It seems like a really good fit to actually accept that one has to hard-cast and focus on DPSing more than the others, and just adopt the healing more around it with OGCDs, and shields to allow you to cast rocks, use aeros, and slap with holy magic.

I say accepting the fact the game is DPS centric is much better than trying to pretend that having a job that doesn't DPS as well via design is going to ever be worthwhile. So long as people think white mage should be the "bigger number heals at the sacrifice of utility and damage" healer in a game in which that is never needed, and in which utility and raw damage are always valued over everything else, then White Mage will forever be poorly designed.

Own the fact the game is about balancing your jobs, which are often at odds with continued DPS, and doing as much damage as possible as fast as possible. Don't try and pretend the game has any place for anything that's just healing and nothing else. If you want the raw numerical healing to be what you want, by design, Final Fantasy 14 is just not that game- nor do I believe it ever really will be just by how most interactions in this game are handled.
More unique fights could make healing more complicated to perform- requiring cooldowns and such, but they simply don't have any room at all for "more healing" in their healers. They all have to be able to survive everything or else they have no point being a healer.

TLDR; the only real way in which to separate healers, while keeping them different and competitive, is the means in which they dps and heal at the same time, not just how they heal. There is no avoiding this.