I feel like it all kind of equals out in the end.
They do pay social security taxes, medicare taxes, state taxes, etc. So implying the poor don't pay taxes on their earnings is misleading. They just don't pay one segment of taxes, which, to be fair, is the highest percentage of tax that needs to be paid.
On the other hand, they have to pay social security taxes on all of their income. Once you go over $137k in taxable income, you no longer pay SSI tax. That's a fact a lot of people don't realize as well.