I would say at the very core of it all is two issues.

First is that as they keep adding new OGCD heals, tank heals and the combat design exclusively centers around short bursts of damage from the boss... Healers keep getting excessively more downtime in fights, but they absolutely refuse to give healers anything to do in this downtime other than spam their one attack spell.

Second is that most healers don't really have any way to really use their Job Gauge in a positive manner... AST cards being a damage boost were always useful, but you need good RNG for Astrodyne and Minor Arcana is a 50/50 to be useful... SCH being the best healer class, it can get rid of extra Aetherflows to do damage so you can always make full use of them, but the attack is a 100 potency one.
The other two are completely shafted as WHM has to overheal with lilies to get to use Afflatus that they had to buff multiple times to not be a DPS loss compared to Glare spam. And SGE has to overheal with Druochole to recover MP.