The TOS Enterprise was supposed to be on a 5 year deep space exploration mission though. They weren't supposed to just drop off and pick up new crew willy-nilly, the crew they start with is supposed to be their whole crew the whole mission.
Not really. Gene Roddenberry's original draft of the narration goes like this:
This is the story of the United Space Ship Enterprise. Assigned a five year patrol of our galaxy, the giant starship visits Earth colonies, regulates commerce, and explores strange new worlds and civilizations. These are its voyages, and its adventures.
Although the narration was eventually revised (by Gene Roddenberry, John D.F. Black, and Bob Justman), the important components to the mission besides exploration (visiting Earth colonies; regulating commerce) were evident on the series from the get-go.
They certainly weren't tied down to the crew at the outset, as illustrated by their regular encounters with other Earth ships, Federation Starbases, and the like.