Monk healing is going to take a stab at it. It'll be using some form of stances and marking to direct DPS to specific groups.

The major difference in handling between tanks/DPS and healers is that the former can learn the keys and focus on the screen, while healers ALSO have to juggle their raid frames. Tanks and DPS (with minor exceptions such as intervention and rescues) only have to watch the boss.

I think there's a lot of opportunity to improve the game by taking out the whack-a-mole (slash untargeted spray). Consider if Disc Priests played by building up various barriers to protect positions and reduce damage taken. Holy Paladins could soak damage into themselves (and focus on healing that away), while Restoration Druids could grow gardens of localized healing. None of these would take raid frames, and (with the exception of the tank healer) healing responsibility would be split between smart placement of effects by healers and DPS taking proper advantage of them.

Personally, if they made everyone identical at tank healing, and then gave us such differentiated approaches, I'd be perfectly happy to be rotated in/out of tank healing vs. area effects based on what's strongest on the particular fight.