Residents elect councillors for their local area based on the policies that the candidates and their parties put forward and the reputation of the candidates.
The group with the most successful candidates runs the council and chooses its leader from among the elected councillors. So that person has the confidence of a majority of the councillors elected by residents. It’s just the same way that the Prime Minister is chosen!
Hence, the leader is also a councillor and has to keep in touch with local people by representing the interests of his or her ward as well as providing leadership of the council.
Under an elected mayoral system, candidates either just decide to put themselves forward or are picked by their party supporters – who could be union members, business people, their paymasters – in a completely closed and unaccountable way. How can that be better than the present system?