I do think that all healing should require hit dice; having a maximum limit on how much you can be healed in a game day might make it easier to stick to encounter guidelines. But at the very least, the Hit Dice should be added to the healing spell or effect, to make it different than just taking a rest. Sure, being able to heal outside of resting is useful, but since casting a spell is already using up a resource, there should be more effect if you're making someone spend two resources (spell slot + hit die).
And yes, as Mark B said, I'd like to see a system where you actually want to cast Cure Wounds to prevent players from going to 0 in the first place. Maybe bring back the Bloodied condition, where a monster might become more dangerous if they bloody a player, thus making it a good idea to stay above half hit points.