we should eliminate the payroll tax

I am not sure I understand your opposition, or maybe I don't understand what you mean by payroll tax. The 12.4% FICA tax pays into the Social Security program. Eliminating the tax would amount to a nice raise, but would require different funding or elimination of Social Security.

There are two interesting features of the tax:

1. There is a cap on how much income is taxed, currently $118,500. Income beyond that level is not taxed. I think this is why you call it "regressive," because very high earners do not pay more than people earning at the limit. But judging by the benefits worksheet, you don't get higher benefits either. No one gets more Social Security benefits by earning above the annual limit. (High-earners were originally supposed to be completely excluded from Social Security.)

So everyone contributes 12.4%, and everyone gets back in proportion to their contributions. Doesn't seem regressive. (I am ignoring the Ponzi flavor of the program and assuming that benefits are fairly calculated: that people who contributed twice as much get back twice as much on average.)

2. Contributions are split between employer and employee. I don't see why this matters to anyone's bottom line. It is sneaky that employees only see half of the total as a deduction on their paychecks, but even if the employer's contribution were printed on the check it wouldn't put more food on the table. Employers don't care how the tax is paid, they only want to know the total cost of keeping another worker employed, so making the entire 12.4% an income tax wouldn't affect salaries.

It appears that only the employee-paid half of FICA is tax deductible, which is strange, but that also applies to everyone equally, and doesn't seem to raise a fairness issue.