This is sorta already how it is in the US. While we obviously don't have universal healthcare, or a national pension, we do have Medicare/Medicaid and Social Security. These are separate than income tax. On your pay stub, it's the FICA ("Federal Insurance Contributions Act") deduction. There is also a smaller separate supplemental Medicare-specific tax. Federal income tax is a separate deduction.

I believe this is how the UK does it as well. You pay income tax, then you pay your NHS fee.

I.e. so it can be properly regressive, right?