Kind of expected this thread to just be here and see nothing but posts from players being madge about the recent FRU clear with no healers and how they want compensation from Yoshi-P and the devs.
Healers shouldn't get mounts the same way that the tanks do when it comes to doing content to earn them. If they want to do it like how you get the Unicorn mount for playing Conjurer to 30 then that's fine. But achievement reward for playing healer in certain content like tank has? They definitely shouldn't.
Lets be blunt here: there are alot of healers out there that are just complete dogshit. Be it from the heal bots that deal no DPS, to DPS healers that do no heals, to those who refuse to heal you because either they/you/the party are toxic/"being toxic", etc etc etc. Healers might be FF14's punching bags when it comes to, well everything pretty much lets face it, but the last thing the healer job needs is for an incentive to be added to reward players for simply "playing healer".
Example: You get a duty with a White mage that only casts Regen on players for healing and tells you and the rest of the party to go fuck yourselves for telling them to do more then use Regen to heal and to deal with it since they only want to deal damage. You all still clear the content. Did the White Mage do their job?
While the answer will obviously vary between players, and there are obviously factors to consider in each scenario, for a good chunk of players: as long as the content gets cleared, how other players' play is completely irrelevant to them since all that matters is that said content they're doing is cleared. Having a healer mount reward for just playing healing would not get people to play healer, it would get people to "play healer".
Plus lets face it, ultimately if they did add a healer mount achievement reward it would just follow suit to how the tank mount achievement is done by most players, aka: finding the lowest level and fastest to clear 8-man raid that counts towards completion of the achievement, getting a party, and just spamming that fight 100/200 times and get the mount quickly and then move on so they don't have to touch the job again.