It’s a legitimate problem. When I heal, I expect 99% of my time to be spent healing, not dpsing. Just like I want to be tanking as a tank, and not kiting.
I think the easiest solution to make both camps happy, is to introduce distinct options for healers. One side should be about weaving personal damage, then there should be double edged sword type buffs you can give your party in a mutually exclusive tree or talent pick.
Let the healer increase damage taken on a DPS, or make the buff a steady ticking source of damage. Increasing the DPS’s damage, but at the cost of needing to be healed more. You could do similar mechanics for tanks, a debuff that cuts into healing taken, but reduces damage taken. Make it a modest net increase, but at the cost of needing more healing taken.
Let healers heal, and find unique ways to encourage that.