The only game other then FF I can speak to in this list is WoW and the healers over there absolutely have healing as the priority. To prove my case I checked Warcraft Logs, and here's what I found
(These are the number of casts)
https://i.imgur.com/Jj8QdoW.png
https://i.imgur.com/FxBul1g.png
Now, WoW has a couple advantages in their healer design that lets them do this. One, there are classes that can heal by doing damage, like Sage but more in depth then just "Heal Kardia target for 130", like Mistweaver Monk or Discipline Priest. Two, the raid comps are more flexible, so you can trade healers out for actual DPS classes if the not a lot of healing is required.