If I'm going to be playing healer, I want to be healing most of the time.

Make boss auto-attacks threatening so that the tank isn't only in danger during tankbusters, a tank shouldn't be able to survive just on their own self-sustain. Also allow the boss to continue auto-attacking while casting other abilities. This would require adjusting tanks a bit but that's a different topic of discussion.
Throw in some frequent, weaker instant cast raid-wides. Every raid-wide doesn't need a telegraphed long cast so you can prepare a shield beforehand, healers already have HoTs and shields they can keep prepped for when these do happen. Bleeds on stronger raid-wides are fine as it forces healers to focus on healing until the debuff ends.

Using a GCD heal should be an important choice on whether or not people survive, not something ignored in favour of doing more DPS. A dead DPS would have a far bigger impact on beating damage checks than a healer's personal DPS.