"You can generally make two generalizations about local governments in the United States: they are local governments and you can’t make any other generalizations about them because everything depends on the State and sometimes a locality’s specific circumstances."
That is good intel though. I thought NY was a home rule state, but is that only for municipalities which have had home rule enabled by legislation? This table doesn't really go into that many specifics: https://en.wikipedia.org/wiki/Home_rule_in_the_United_States
One weird NYC thing is that it is the only city I am aware of in the US that sits above the county level. The five boroughs are the five counties and mostly used for administrative management, since four of the boroughs are million plus cities in and of themselves; but the boroughs have no lawmaking powers.
In general, counties are pretty weak in New York.