This is another place for 5e to shine with options that may or may not be included in the game at a group or DM's discretion.
I believe clerics (at least those of beneficent gods) should remain the best, default "healers."
Druids and Bards can use "healing magic" but not as much/as well as a cleric. I'm fine with that.
Paladins' "laying hands" healing is pretty "iconic"/original to the class. So I'd make sure that stayed in.
Necromancers would be the only "Arcane" type I would consider allowing healing...and/but that, only by "draining off the life energy" of some other creature...even if it is themselves...or a tree...or something.
But anyone should be able to take a "healing" skill or "field medic" proficiency or (to use the coalescing 5e vernacular) "Theme" or something to that effect. Everyone should be able (with the appropriate skill/theme) to aid companions, in or out of combat.
If I don't have an enemy barreling down on me, I should be able to forego attacking to staunch the bleeding of my fallen ally. Or as we used to do it, halt the descent into neg's before you got to -10/dead. Not necessarily "add" HP, but at least halt or slow "loosing" HP...and/or only add 1/improve natural healing very minimally.
A "Healing Surge" by any other name twould work much better. A "Second Wind" or "Temporary HPs in combat" or something for those who like the "inspiring" support kind of classes...or even certain other classes (Barbarians being able to "boost" their own HP in their rage, or some such...improve their natural healing/heal faster?) would work fine. But not "POOF! You [anybody of any class] have 10 HP permanently back!"
Healing potions should be available...what "easily" means, I'm not sure. But going into the Temple of the Goddess of Healing should, yes, have potions or Scrolls of "Cure Light" for sale...er...I mean..."donation"...(higher level curatives would probably be kept in the temple vaults, for the faithful only, or not scribed into scrolls at all to maintain the priests' position of importance).
...or the best Herbalist or Alchemist in the land? Sure. Healing potions should be available in a number of ways...and, I would add, with varying effectiveness (dice rolls to determine how many HP you get back. I'd even allow, like, d4, d6, d10...hell, every "d" if you wanted, different types of healing potions.)
"Cure Light Wounds" wands on every street corner ought to be a thing of the past. Crafting their own? Ok...when/if they get really "high" (I'd say, 10+) level...if magic item crafting is part of the game at all.
Finding/discovering a "long lost Staff of Healing" that does all kindsa curative things (with limited uses/charges, of course)? Definitely. Items that allow regeneration, or protect from poison, or what have you. Yes, but really rare...as magic items of any kind ought to be in general.
--SD