Here are my half-baked thoughts on how to differentiate the healers while making them more fun and engaging:
WHM: Nuke all direct-healing oGCDs except for Bene. Add an oGCD that is Wildfire, but for healing - when it goes off, the raid is healed for some percentage of the amount healed on the original target. Add a short-cooldown oGCD that converts all remaining regen ticks on the target into an immediate direct heal for the same amount. Make MP management part of the class. Gets a "Euphoria" mechanic proposed elsewhere, where overhealing a target to 125% HP will give the target a damage buff and place a DoT on them. This makes all of WHM's healing output useful, and is an engaging way of buffing rDPS. A good WHM will be able to keep Euphoria rolling on the DPSers while healing through the resulting DoT and the fight damage.
SCH: Eos's DPS pet. Aetherflow is now built by dealing damage with DoTs and a proper DPS rotation, instead of being given by an oGCD. Meltdown is a DoT that boosts damage of the SCH's spells against the target. Drain will heal the closest party member to the target for a percentage of its damage (shamelessly lifted from Atonement in WoW). Adlo/Succor become Aetherflow heals and get stronger (but can no longer crit). Instead of healing, Fey Union now burns fairy gauge to give a damage boost to the target, like a dance partner or DRG tether.This makes SCH's buffing more engaging than just smashing Chain Strategem every two minutes. A good SCH will keep their DPS up so they can generate Aetherflow stacks to deal with fight damage, fill the fairy gauge, and Fey Union during burst phases.
AST: Go all-in on the time magery. Instead of shields, Noct-AST turns Aspected spells into delayed heals. Aspected Benefic/Helios under Noct puts a buff on the target that heals for big potency when it expires. Di-AST regens no longer stack with WHM regens, because allowing four regens on one target is stupid. Aspected Malefic is a DoT (Di) or a delayed damage (Noct). ty_taurus's brilliant card system replaces that...thing that AST has now. Time Dilation returns as a GCD that is the bread and butter of AST's gameplay now. Aspected spells' potency will snowball, increasing with each tick up to some limit, as long as the effect remains on the target without interruption. This limit will NOT be reached naturally during the short duration of an Aspected spell. Time Dilation extends all AST-supplied effects on the target for two or three more ticks. This keeps Di-AST regens and DoTs ticking away at their max output, and Noct-AST delayed spells gain extra potency with each tick of the buff when they eventually expire. A good AST will be able to keep damage effects rolling on the boss, big heals rolling on the tank or party, and juggle buffs on a couple of players.