Not focusing the game on in-combat healing is key to fast combats, I would say.
Nod. There's a difference between focusing on it and allowing it to be effective. Fast combat means there'll be some high-volume damage-trading going on over a fairly small number of rounds, and thus small number of random resolutions, small sample-size = higher variance, so an even slightly unlucky PC could go down early and miss virtually the whole fight, or worse, simply be killed outright (unlikely much above 1st level), or worse, start a party-wide death-spiral for want of that character's contributions in the combat. So for campaigns to consist of 6-8 'fast combats' per day, and a couple days per level and hundreds of round in a campaign, without multiple TPKs being inevitable, those combats have to be tuned to being almost trivially easy and/or the party has to have a 'come back' mechanism.
In 5e, obviously, encounter guidelines do err on the side of being 'too easy,' but in D&D in general, in-combat healing/restoring has been that come-back mechanism. 5e in-combat healing is not quite up to the challenge, so players try to use it most efficiently-which means waiting for an ally to drop, then nudging him back to consciousness as a bonus action to take his turn without sacrificing yours. Which, if you do it more than once, becomes whack-a-mole.
As usual, I'm verbose, but ultimately in agreement: fast combat can't mean focusing on healing. But, it must also mean having fairly potent healing available for 'emergencies.' Fast combat can also mean that combats often feel 'too easy,' so challenging-feeling ones may well call for some big in-combat healing.
But (very) generally speaking, a game can decide to offer much/cheap/fast healing, or it can decide to not do that. The latter decision will generally result in faster combats.
Sufficiently fast healing on one side of the combat would be quite compatible with fast combat, if it lets the healer belt out his usual DPR, and keeps his allies belting out theirs, instead of laying on the ground making death saves.
Healing Word doesn't quite pull that off this ed, because it's not strong enough, and leaves the healer only a fairly minor single melee attack or meh cantrip with which to chip away at the opposition.
So the key is to offer combat healing in strict moderation.
Like, IDK, 2 healing words / encounter? ;P
Of course, the price that game pays is that it no longer supports the "holy trinity" model of game combat; the tank, the damage dealer, and the healer.
Fast combat as too high a priority does distort the game towards an all-damage-dealer pathology, sure.
But you could argue that D&D doesn't do that anyway (and that it's attempt at it, 4E, wasn't successful, though possibly because of entirely unrelated factors).
That'd be a tough argument. AFAICT, D&D has not only always done that (had meat-shield, DPR, healer, & magic-slingers integrated into a model of combat), it originated it, and video games, CRPGs, & MMOs merely adopted it. (And, yes, 4e did it in a particularly formal, balanced, and clear way that worked more cleanly than in prior or subsequent eds, but was subjected to edition warring.)
Anyway - I feel 5E has succeeded in finding a "feel-good" spot.
Even though you complain about it being 'too easy?'
If you really want combat healing, you can have it (=Life Clerics, and assorted other specialized builds). Otherwise, the game works fine anyway.
There are /4/ classes with significant in-combat healing-others ability, and a 5th that can have a little, and 6th that can heal itself. So, if you really don't want it, you'd have to toss two classes entirely, and willfully avoid knowing/prepping healing spells with 4 more.
Which brings me back to my original point. To anyone feeling their Clerics are forced into spending too much effort on healing others, before you add house-rules, do try out the healer-less party for an adventure or two. Chances are that once your players are weaned off the healing teat, they will come to the same realization me and others have already realized
Handy old-school table-policy for healer-less parties: first one to die rolls a cleric.
