Some cities have both - an elected mayor to perform the ceremonial duties and then a city manager appointed by the city council to conduct business.
Conventional wisdom is that the city manager model is better but obviously not in this case.
My older, and now deceased, brother spent his career as city manager to various little towns. The problem was that the city council would change, and the new majority would expect the city manager to kill their asses. When he refused, they'd fire him. He lasted, on average, about 5 years per city while the average was about 3 years for most people.
Some cities have both - an elected mayor to perform the ceremonial duties and then a city manager appointed by the city council to conduct business.
Correction, the mayor in a Council/manager model is essentialy the chair of the board leading policy development and budget approval nad oversight.
The council hires the city manager who then hires and manages all staff (except sometimes a council has direct hire of a couple of people like city clerk and corporation counsel as is done in Portland Maine).
This is when done right.