There was no need to go to the Metropolis police department. Superman was in that city, and he practically handled the work of every police department, fire department, National Guard unit, and more. As a result, the Metropolis police department was famously incompetent—almost a model showroom for existentialism.
Simply put, as long as the city had a police department that could mediate neighborhood disputes, write tickets, and tow cars, that was enough. City hall never expected them to do anything else.
If the Gotham Police Department's combat strength was a ten, then theirs was a two or three. Whenever something major happened, they did not even need to call for backup on the walkie-talkie. All they had to do was shout at the sky, "Superman, save us!"
Over time, who would still train seriously? They might as well get up every morning and practice their voices, making sure they could cry for help more sweetly and melodiously.
As for how such matters should be handled, Batwoman and Green Arrow belonged to one camp. They believed Superheroes should not get too involved in government affairs. They only needed to deal with unnatural incidents and powerful enemies that ordinary officers could not handle. For example, if Penguin was smuggling military weapons, Batwoman had to get involved, because he was definitely plotting something big.
But if some ordinary nobody was smuggling military weapons? Just call Gordon and let the police handle it.
Only then could people realize that they had to rely on their own efforts to survive and live well, instead of relying on Superheroes whenever anything happened. What if the Superheroes died one day? Who would people rely on then? Themselves, of course!
Superman and Barry belonged to the other camp. Whenever they saw something that needed doing, they wanted to