Well, local governments (cities and towns) also have expenses -- police, fire departments, trash collection, water and sewage, roads, public works. Schools are partially funded locally. That has to be paid for.

It's theoretically possible for a local government to levy an income tax, but a lot would need to change -- much more than just changing tax rates. Employers and banks report income to the federal government (and states, I suppose, but I live and work in Texas so I don't know much about that). They would have to report that information to towns and cities too. There's also the problem of granularity -- how does an employer or bank know where someone actually lives? If you have a P.O. box in a town, do you have to pay taxes in that town? If you work in a different municipality (not uncommon!), do you have to pay taxes there too? If you have a home in one town, work in another, but spend most of your free time hanging out in a third, are you completely off the hook for supporting the third town?

You could have the federal government collect all the money and then allocate it to state and local governments, but that's a massive change in how American society works, and I'm not sure it's any less complex in the end. Some of the complexity in the tax code (e.g. different levels of capital gains tax) is a policy choice, but some of it reflects the complexity of the real world.

> It's theoretically possible for a local government to levy an income tax,

“Theoretically possible” in that thousands of local jurisdictions, among about 1/3 of US states, already do either income or payroll taxes or both.