One of the reasons to have a separate director and stage manager, even in the running of the show, is binocular vision.
When putting a show together having a director, various designers, stage managers and actors all bring separate points of view to make the show strong coming from different points of view.
During a run, when a stage manager watches the show on a daily basis, it is actually harder to maintain a show then a director who comes and sees a show once a week.
Look at a flip book and compare a picture on page 6 and page 7. It's hard to notice those differences, but look at page 6 and page 13 and you should have an easier time to see what's going on.
There is also something about a show opening and leaving behind the director and the rehearsal process and moving on to a new phase. I have found no matter how nice the rehearsal process is, most actors are thrilled to move on to performance phase, and leave behind the rehearsal phase. I'd worry that what ever tension may exist between director and the actors during the rehearsal process would carry over into the run of the show - which could be very problematic.