A while back, I was speaking to a young rabbi who was entering his first pulpit, and he mentioned that in a shul he would have the freedom and flexibility to do that which he thought was most important for the community, and to do at the time that he felt would work best for thim and for the community. That reminded me of an important lesson regarding the synagogue rabbinate, as well as life in general: Don't confuse limited autonomy for total freedom.
It is true that shuls tend to trust their rabbis to make their own schedules; the rabbi decides when to visit people in the hospital and when to prepare shiurim, how much time to spend on counseling and administration and teaching and tzedakah distribution, and whether the shul needs another shiur or another chesed program. However, the rabbi who mistakes this brand of autonomy for total freedom is, in my opinion, making a significant error.
The shul rabbi's autonomy is like that of any contractor – the board wants a healthy community, and trusts the rabbi to decide how best to do that. However, the shul has a vision of what a healthy community looks like, and the rabbi who ignores their vision in favour of his own does so at his own peril. [Note: the wise rabbi will openly and honestly share his vision of "healthy community" when interviewed, and the search committee should vote for a rabbi whose vision matches that of the shul.]
If the shul wants a community in which members regularly consult with the rabbi about their personal troubles or schmooze with the rabbi at the kiddush, then the rabbi had better make sure to provide that.
If the shul wants a community in which the rabbi teaches a shiur for every group of three Jews who want it, then the rabbi had better make sure to provide that.
If the shul wants a community in which the rabbi is a regular contributor to the Op-Ed columns of the local newspaper and a bridge-builder to other sectors of society, then the rabbi had better make sure to provide that.
Of course, there is ample opportunity for the rabbi to sell his vision, and if the community responds well, then that may come to be the community's vision. And the sensitive rabbi is open to learning and evolving, and adapting his vision to the lessons he picks up in the community. The "healthy community" vison may well be a moving target, and both parties can/should shift and grow.