As much fun as this would be, I'm worried this would make them automatically the best healer- and if they aren't they'd be not that great I'd imagine.
Here's what I mean.
Healing in 14 is managing to keep everyone alive through mechanics, and raising them if need be, to allow the most damage towards the boss as possible
it may be a round about way, such as esuna-ing something so you can use more casts towards DPS skills instead of more cures down the line, so on and so forth.
But if a job has it's main gimmick be that it does both at the same time- not even at all times, but that's it's main source of healing, it's not doing both at once.
At that point, we'd have to ask "how much damage" and there in lies the problem. Either they don't do as much DPS as anyone else can by them balancing between doing their role and murdering the bad-guys and they're worthless by offering nothing else. Or they do more damage than SCH/WHM does or, good lord forbid, even more than the raid-wide damage AST gives via cards and they're just seen as the one mandatory job- and we're just back to square one at that point.
When the healer's gimmick is 'not having to balance between the role and DPS' that ironically forgoes the entire role of being a healer in ff14. It'd be like a tank had it's emnity combo also be what you need to only spam for highest DPS- or it having a stance that gives both 10% damage reduced and 10% damage given increased.
The roles all have jobs to do that go against killing things as fast and effectively as possible, and by putting a healer that does both in the game, either they do it effectively and simply become the automatic best, or don't do it as effectively as others that have to balance them and feel pointless.
It's a shame because, honestly it sounds like damn good fun!
Personally, I would instead turn this from a job and into traits to give to WHM in a more indirect manner. Such as 'every X consecutive stones that hit the enemy give you an instant cast cure1' or 'every Y Aero damage ticks reduces tetras CD by Z seconds' or something along those lines.
I wont act like I know the balance, but I think taking that idea and incorporating it into WHM in a more indirect manner would be the better option. That said I'd love to be wrong and for the job of a 'DPS-to-heal' type to somehow work perfectly fine, but with how the game simply handles healers and their path to optimization, I just don't see it unfortunately.