Hence they vote primarily for policies and the duty of those they elect is to do their will by sorting out the details while carrying out their wishes. The directly elected mayor will tend to stand not on policies but personality. As such there is a danger the mayor will be someone above politics, a man trying to act as a saviour rather than a servant to do the voters bidding. The personality of the mayor will be crucial. I am not confident in judging the personality of people I know - to judge someone, reliably, by their media personality is next to impossible.
Parties have to stand on their record.
Voters can punish parties that seem to have a culture of sleaze and so parties will tend to compete in keeping potentially corrupt members in line. No one can be an all rounder. In areas that are not the prime interest of the mayor these Sir Humphreys will call the shots and there supposed neutrality will just be an insulation against unaccountability. The strength of the system of sub committees that is normal in local government is that each sub-committee will be composed of those primarily interested in the are of concern of that committee. If there must be a mayor then they should be elected by a Condorcet criterion method.