Look at the math... and remember that dungeon crawling is at least partly about resource management for many.
Stabilizing someone without a healing kit: DC10 Wis (Medicine). If pretty typical, this will be a +0 or +1, +2 or +3 for a starting cleric, monk, or paladin. So, 55-70% chance. Add medicine first level, and it goes to 65-80% at first. So your healer cleric becomes MUCH more duration-focused with Medicine skill instead of Healing Kits.
By 5th level, the healer types are likely to be +4 stat, +3 PB, and so 90%. ANd they may be +5 stat bonus...
Rogues as healers using expertise can hit automatic right quick.
Level 1, assuming the standard array's 15 in Dex, 14 in Wis, a rogue-based medic is doing +6, or 85%, 90% if human. At fifth level, he's got a +6 expertise... and possibly a 16 Wis, giving a +9... which, being an ability check rather than an attack, isn't subject to autofail on a 1. At 5th level, a rogue healer can be an auto-make without kit.
From a resource management point of view, it's an excellent choice of skill in the long run.
D&D is the topic at hand, and D&D is a roleplaying game.
When talking about a roleplaying game, roleplaying is not your point, is that correct?
Not everyone plays RPGs as narrative focused story-games. For some, they are a subset of story-driven boardgames. For others, narrative wargames.
D&D has ALWAYS supported both the character scale boardgame mode, and almost always the narrative wargame mode. OE was, officially, an add-on to Chainmail miniatures, not the other way around. AD&D 1E and 2E had Battlesystem as an add-on and/or standalone.