A lot of this, of course, is just a matter of dramatic convenience. Giving Kirk a specific crisis to deal with or a short-term mission to carry out gets the plot rolling much faster, and gives an episode (or novel) more of a sense of urgency, than if he's just exploring some random solar system for the heck of it and something unexpected happens.
Which is not to say that you can't get a decent plot out of a purely exploratory mission, but the show was smart in not defining the Enterprise's mission too narrowly. Giving it a wide range of assignments, including military ones, opens things up to a greater variety of plots. You can do war stories, disaster-relief stories, espionage stories, first-contact-stories, morality plays, surreal mind games, horror stories, etc.