The federal income tax as well as state and local taxes would still be in place.
Right, but since it’s the payroll tax (FICA) that funds Social Security, eliminating the payroll tax would scrap the main source of revenue for Social Security.
It’s been a while since I saw a pay stub, but if I remember, there were separate deductions for federal income tax, state income tax and fica. The former two paid solely by the employee and the latter paid 50/50 employee/employer.