I think most of us wouldn't, if they could actually find a way to distinguish them from one another, but as it stands it feels like the split was an experiment that hasn't gone well, like the SB WHM 'let's try making it have really strong healing as it's identity' experiment that lead to WHM being the black sheep of the game. The point I try to make here is that, despite having fairly similar kits in terms of mitigation options, Tanks feel 'different enough' from one another. GNB feels different to PLD, PLD feels different to WAR. This is because their damage rotations are different, which is the majority of the buttons they will be pressing in content. As such, we can make the healers feel 'different to play' via a combination of 'the DPS rotations feel different to one another', and some of the OGCDs feeling different. As an example, healing on WHM vs AST feels very different to me, as WHM is very GCD heal reliant, with systems to refund the damage lost from that GCD healing, whereas AST relies on 1min OGCDs a lot, with CU CO and Star being the primary examples. SCH and SGE are very similar to each other in terms of OGCDs which is kinda bleh, so setting them apart in gameplay by making their damage rotations the point of difference would be the way to go.
Again, for example's sake, let's assume the ideas I posted in this thread are all adopted into the game (almost impossible, but I can dream). The four healers would feel different to each other because WHM would have a very burst based damage rotation, SCH would be juggling DOTs, SGE would be managing procs like it's playing DDR, and AST would be lining up it's delayed damage 'Omen' spells to all strike the enemy at once. The fact that Lustrate and Druochole, or Indomitability and Ixochole, are exactly the same potency and range, wouldn't cross anyone's mind, because you only use those once per 30s. The damage rotation is front and center, so that's the part we should focus on when we want to make things feel different from one another.