I would think that by the 23rd century a Doctor would have a breadth of knowledge about other humanoid species.
It seems to me that by the 23rd century, an MD would have even narrower
knowledge than today. Medicine is advancing so fast that it's basically impossible to be a general practitioner today; it might well take a minimum of three people with doctorates to adequately cover even Homo sapiens
The only reason I think McCoy might be expected to know something about Vulcans is Spock being aboard. Apparently, the Vulcan is the only nonhuman on that ship - so Starfleet would do well to pick a doctor with at least some proficiency in handling Vulcans. But for all we know, they did, and that doctor was M'Benga; due to his rare specialization, he in turn was an inferior choice for treating the 429 humans, or for some other reason (lack of service years, say) couldn't become the CMO.
A smart organization might have given up on human doctors altogether, and only enrolled Denobulans or other species capable of coping with broader skill sets.