You know, if by default no single source of healing did very much, you could "boost" certain sources or pile on additional sources in your campaign to set the level of healing that you wanted. Maybe clerical magic does a little, herbs do a little, healing kits do a little, rest does a little, etc.
Then in those subset of games where a character wants to be the dedicated healer, he can play a cleric that has the skills and background necessary to make kits and herbs and potions to supplement his magic. That even fits some fantasy lit. OTOH, if you have a group were no one wants to do that, then the cleric is doing his part by having a few spells, and he can devote the rest of his character resources to being that war cleric or trickster priest or whatever. Then if the fighter, rogue, and wizard want more healing in the party, they can take some of those other options.
That doesn't solve the issue of an expected level of healing and what happens when it's not met or exceeded, but that's a thornier issue that requires some better levers than a running hit point total, anyway.