Every edition of D&D has had issues with healing. 4e became especially bad when healers became so potent as to keep the party alive almost indefinitely. This lead to many complaints that 4e combat was too slow and too easy. A simple solution to this would be to assume players have no (or very little in combat healing). Then clerics will not be required and the game will be a challenge as there will be very few things that can extend your groups endurance during each encounter. Clerics who have to use an action to heal (such as with cure light wouds) will then have to make a meaningful choice, instead of being relegated to band aid duty. Most healing can then be relegated to first aid and short rests completely removing the need for magical healing if the group so chooses.
My 5e Homebrew Material