Don't try to make sense of the US tax system$
(Not that yours makes sense either, but at least you have National Health and university education.) More or less accurate off the top of me head (New York resident):
Federal Income Tax - pays for the operations of Fed Govt, especially military and Medicare (retiree health), and interest on funds borrowed to cover deficit spending.
Alternative minimum tax (part of Fed) - originally intended to get rich people to pay at least some tax, now overcharges quite a few middle class people. Attempts are being made to fix this.
Payroll tax - pays for current Social Security retirees (matched by employer). Currently paid only on earnings below about $100k. This is illogical, used to be worse.
State Income Tax - optional by State, pays for operations of State govts, including State Universities (not free but less costly than Private), Medicaid (some health care for the poor), roads/infrastructure, some grants to municipalities, and other obligations including those matched or subsidised by Fed Govt.
Real Estate taxes - generally used to pay for public (free) schools. Areas with large average holdings pay for fewer students than more densely populated areas, thus per-pupil funds are less where people are poorer, in general. There is an active movement to change this illogical funding.
Sales Taxes - like VAT, inherently regressive, as lower income people spend virtually all their income (paying this tax) while higher income people spend a much smaller proportion of their income on taxable items. There is no sales tax on stocks and bonds, and the increase in value is taxed at the capital gains rate, generally less than the personal tax rate. Sales taxes are pretty much the only taxes that are easily controlled by the State and Local governments, so they are popular with Governors.