To slightly correct, a character with the Healer feat can expend a use of the healer's kit to bring unconscious creatures up to 1 hp or heal them 1d6+4+HD (once per short rest, per character). Nothing restricts a medicine kit for sole usage by the Cleric class, and similarly any character can take the Healer feat to get these bonuses while using them.
But this also means that the ability to bring unconscious creatures up to 1 hp via a healer's kit stabilization requires an entire feat, and it's not something given for free just for choosing a class. So the fact that all characters can do it for free in BG3 is even more powerful than RAW than OP implies.
But yeah, +1. I dislike the whack-a-mole situations encouraged by 5e's "it's more effective to heal unconscious characters back into the fight than to heal them while they're still conscious." And I dislike BG3's exacerbation of this phenomenon even more.
