I'm sure it is somewhat unusual, doesn't mean it's impossible. Janeway was a scientist, so was Spock. Both became starship captains. If that is normal I don't see a medical doctor becoming a captain being much of a stretch.Neither is a field that lends itself well to part-time practice. Why would you want a surgeon who had spent part of her career learning to be ship captain, or vice versa, rather than one who had concentrated their entire career on one or the other?
As for why one might want a doctor captain, I presume that a purpose of a hospital ship is to render medical assistance to distant planets or stranded ships in case of a disaster, outbreak or other medical emergency. As such, many decisions the commanding officer needs to make, are medical decision. They must be able to asses the priority level of an emergency and assign best possible procedures. And if you say that a non-medically trained captain can have a CMO to do such decision for them, sure, that is possible, but in that case the captain becomes a glorified taxi driver.
Similar arguments of course apply for assigning a captain with scientific background to a vessel which main purpose is scientific research.
McCoy is not, but Crusher is. Crusher's primary job on the ship is to be a Doctor, just like Scotty's primary job is to be an engineer. And both are qualified and capable of commanding the ship if so needed. I don't find this particularly weird.Those are their jobs on the ship (though I admit Spock's two hats is odd), but they are still line officers. McCoy is not.
It is possible and even likely that it is indeed decentralised and only the main facility is in San Francisco. When the characters refer to the Academy, they may be referring the institution as a whole rather than some specific location.Why? That seems like a big bottleneck if, say, things had to be scaled up in wartime. Not to mention if it were obliterated by a V'ger etc. Why not have it decentralized with the equivalent of ROTC, OCS, merchant reserve, boot camps etc., or even multiple Academies? It just seems imagination-constricted, like "We know these Starfleet characters went to the Academy, so they all must have."
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