The Five Year Mission is, in the title sequence, said to be about them "seeking out new life and new civilizations" and going where "where no has gone before".
Going where no man has gone before, means going away from charted space and not going back into charted space. It's about going deeper and deeper into uncharted space and not leaving until the 5 years are up.
Going to colony worlds, starbases, etc, is them going where other men had gone before, and thus violates the title's premise.
There is a difference between uncharted space and unvisited space. For all we know there could be a dozen systems within a few weeks travel of say stabase 11 that had never been visited by a manned ship.
There is also of course the simple fact that's it's primary mission could have been the whole seek out new life......
but that wouldn't mean that Starfleet couldn't give it another mission i.e. Disaster relief.
And no it doesn't violate the premise, simple because all it states is that the Enterprise is on a five year mission to seek out new life and new civilisations....
Back in the days of sail, I might be mistaken but didn't those ships that where exploring the oceans of Earth sometimes call in at known places?