If you're too poor to afford health care, you'll go through a local government to get care from a local provider, with the state providing a partial rebate to local government. Most of the government services you use will be local, while the bulk of the taxes you'll be paying will go to the state. The state will even get a piece of your insurance premiums, another instant and exclusive revenue source. And if you need a professional license, the state will charge you for that privilege too. Cities collect another business license fee from all businesses, and a variety of other fees.
Counties get a few more, but they are, with few exceptions, comparatively chump change.
Counties and cities do have some limited exclusive revenue fees they can impose. The county collects a sewer fee and the city gets water bills not only from those within the city limits, but many outside of it. In both instances, the funds collected must be used for the purposes they were collected and cannot be diverted for other things. Some creative bookkeeping has been used, but by and large both jurisdictions spend sewer-user fees on sewer expenses and water bills on water expenses. Tells you a lot about the priorities of our state Legislature.