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.
We pay a "national insurance" tax in addition to income tax, but this is mainly to cover the state pension from what I understand and maternity allowance etc.
As I understand it, NHS comes from general taxation.
Obviously there is no actual "fee" to use the NHS at the point of use - it is all free apart from prescriptions which are the same price regardless of what you get (and you might get it for free anyway depending on your circumstances)
