A comment was made on gosanangelo.com that deserves much further explanation: "(When it comes to appointing the police chief) "Hiring" a President to lead the military, would amount to the same thing." Sounds good, but the comparison is seriously flawed.
The Presidents powers as commander in chief are spelled out in Article II, section 2 of the Constitution. There he is given control over all military branches, including the militias when called into the actual Service of the United States. He also makes treaties and agreements, receives and appoints ambassadors, and appoints most other federal officers and judges with the advice and consent of the Congress. In Article I he is given the power and duty to approve laws, including the budget passed by congress (with the exception of those laws where his veto is overridden.) He is given the power and duty to enforce those laws.
At the city level, equivalent powers are distributed between the Mayor, City Manager, and City Council. The Mayor is in charge of all public safety services, including law enforcement, in an emergency because he is by ordinance and state law the Emergency Management Director. The City Manager, by city charter and state law, shall see that the laws and ordinances of the city are enforced. The city manager develops the budget, which is then approved by the City Council, including the Mayor. The number of city employees, including police officers, is set and approved by the City Council. All agreements with other government entities are approved by the city council and signed by the Mayor or City Manager as appropriate. All contracts are subject to council approval. The police chief can't negotiate or sign a contract without prior council approval for that specific contract.
The Police Chief is really closer to the Chief of Staff of the Army. He controls the largest, but not the only, law enforcement organization the city has. He has little say on the laws he must enforce. He has no control over how much money his department gets. He doesn't control the number of officer positions available. He doesn't set the pay rate for his officers, that is done by the city council. He doesn't set the promotion requirements, that is done by the civil service commission. The only positions he can promote to are the assistant chiefs positions. If he fires or suspends an officer, that can be overridden by the civil service commission. He is in very much the same position as an Army General who is given a command and told to go carry out the mission that the President directs him to.
With the Police Chief having only daily operational control over a part of the overall law enforcement or public safety resources of the city, it is hard to see his job as being in any way comparable to the Presidents. There have been countries that have tried electing their generals in the past, but they have all abandoned that as a bad idea. If you are looking for the local equivalent to the President, that would the Mayor.