Every problem with the current iteration of healers.

[Acidblood]

For me the three biggest issues with the current healer design are:

Lack of reward for good gameplay
Examples of 'reward' are DRG's Raiden Thrust, BLM's Polyglot gauge, or 4.X SCH's Quickened Aetherflow... just something that needs to be thought about / paid attention to and gives you something nice in return. WHM lilies are not an example of this as they require neither interaction nor attention.

Lack of interaction within the kit
Good examples of 'interaction' are 5.X WHM Plenary Indulgence and Temperance (which really should be an upgrade from Divine Seal) ... however perhaps the worst example is WHM lilies; they are essentially instant Cure II and instant Medica, so why do we need two extra buttons? It would feel better if lilies just made Cure / Cure II and Medica / Medica II instant cast, which would not only introduce some choice but free up buttons for something more interesting, like Aero 3? PvP Protect? Stoneskin? Cleric Stance? Divine Seal? an oGCD damage spell?.

Lack of general thought required
Due to the first two issues it feels like Healer gameplay has become very stale, just spam (your now limited) DPS until you need to use a healing cooldown (of which there is now an overabundance)... no thought, not much choice, no real ability to optimise, just play like you are a bot (and hope someone gets hit by something they shouldn't to break up the monotony) :/
And of course the new AST cards are the worst example of this, say what you will about the old cards but at least they required some thought.