It shows that they are genre savvy. They know that there is at least a 25% chance of some bizarre medical emergency occurring on each mission and that the doctor as a key character is functionally indestructible 
But that doesn’t explain all the trips to uninhabited or federation outposts where the captain, science officer and doctor HAD to be in the landing party.
Or why a-horse? Why not a-Ford? But I was only fooling. They’re going to build a tunnel in the morning.
(Sorry, but for those of you who may not know, it’s an old Marx Brothers routine. Thanks, Jayjay!)
Radishes
As for this point, it occurred to me the same role could be filled in FP and probably in other fiction by the cook. I noticed in FP the cook was the only one who didn’t wear the starship uniform, until he was being reprimanded.
But others make a very good point, in that the doctor was more on the level of the captain, and could be counted on to be a sounding board since he didn’t know all the details of the ship operation. However, the doc on “The Love Boat” was there for the romantic angle and some lame comedy.
I haven’t read Patrick O’brien’s books, so I am surprised there was a doc there, too. But it should be considered that FP could very well have a basis for his doc, if he, as a child, had seen it.
Radishes
I concur: “A Commanding Officer doesn’t need intelligence, just a good loud voice.”
That’s just the most obvious example. Morbius rides Commander J.J. Adams throughout the movie. I’m surprised Adams didn’t pop him one.
Note, though, that the Captain’s intelligence did measure out as being significantly above human average. It’s only in comparison to the Doctor, and, of course, Morbius, that he’s unintelligent.
And cooks are relatable, but they’re also low-class. The cook in Forbidden Planet was the Trinculo equivalent, more of a fool than a hero.
Or have a Formidable Sidearm Cleaning Accident.