Just looking at the healer list, to start

Using cooldowns- that would be interesting to hear from Square -i.e.- what their perspective is
Using damage skills, while also healing -- already part of what every healer should absorb through the role quests, so meh
Using better/upgraded spells instead of Cure/Physick/Benefic/Diagnosis ;> fair enough, even better if the skills were upgraded through a redesign
Mana management, as a addendum of the point above -> not something I would have identified as a teaching point (as it isn't a pain point), but can't hurt
The different philosophies between heal (reactive) and shield (proactive) healers -> honestly best of luck with this one, I don't think Square has a good idea themselves
Using Esuna (DT healer role-quest does this but it needs to be a LOT sooner) -> DT hammers healers over the head so many times it's worse than condescending, not to mention boring for a skill that ranges from must-use to use-but-no-one -will-die - but WILL get rotation/ movement affected (slow- heavy) - probably ignore