If your premise has characters facing danger somewhat frequently, it makes sense to have someone who can heal around.

Also, with so many unknown space-originating ailments, it's creepy to hear a doctor of high rank say "I don't know what this is or how to fix it." It adds to the stakes of the show.

Stories often have an author-as-character, where the writer(s) use a certain wise character to voice their own opinions.

TOS just merged these ideas into one character.

Did the ORIGINAL series set the precedent for the Doctor character? Hmm, let me think about that one. Yes, I believe it may have.

MAS*H loved the idea, they took it and ran with it.

Specialised crew members capable of providing medical care have been a feature of military vessels for at least two thousand years. — wikipedia

Adaptations of Sherlock Holmes? Doc Watson is the narrator of their stories and sometimes (not very often though) shows his medical knowledge as an ex military surgeon.

I mean, you have all these people, unique, varied and dangerous situations... The role of the doctor would be significant.

If I was in charge, I'd maintain consistent rapport, if not relationship, with them.

I dunno if it was a trend but I'm told that Doc Walton was pretty well-known.