Some random thoughts:
I've recently been running a Dragonlance campaign using the RuneQuest rules. Now, DL (at the start) has *no* magical healing until the healing goddess Mishakal is returned. RQ as a system is quite gritty - characters can be taken out with one good sword blow, theoretically. More often than not they will be injured in a limb and out out of action for about a week. Natural healing will heal a light injury in about a week, with two or more for anything worse.
It was a different experience. It didn't deter the players from combat much, but has seemed to have instilled three principles in them - 1) armour is your friend, 2) get the first strike and make it count, 3) be very careful with big monsters.
Second point:
In Monte Cooke's Arcana Unearthed, magical healing is slightly less common. Instead of CLW, the first level spell equivalent is Lesser Transfer Wounds, which takes a Full Round to cast, heals 1d10+CL but transfers half that to the caster as nonlethal damage. The next spell, Lesser Battle Healing, is a Standard Action casting but cures slightly less (1D6+CL, I think), but doesn't affect the caster. It's 3rd level.
On the other hand, *any* spellcaster can prepare these spells, further there are a few classes with class ability healing and a few feats that help. The flip-side of caster flexibility is that you don't have a dedicated Band Aid mage, so spell slots used to heal are spell slots unavailable for buffs 'n' blasting. It plays marginally differently to D&D, but not massively so.
Combined point - I've tried various systems with and without extensive magical healing, and I think (unless you want really grim like Call of Cthulhu) it's better to have some sort of unrealistic recovery time otherwise your characters spend a lot of time in hospital! (This, of course, is a boon in CoC because that's when the evil cultists come back to finish off the hapless investigator, or better still send a beasty from Beyond).