Not really into math are you?
To answer your question, yes, Federal employees pay income taxes. While it’s odd that we collect our pay from taxpayers and then give some right back as taxpayers, that’s how it is.
I think the OP’s point is that it’s the same money being moved back and forth.
They could just as easily pay the Federal worker $15 instead of $20, and say he owes no income tax, and not bother to shuffle the money back and forth. IOW, there’s no real revenue there.
Whereas the private employee provides revenue, “new” money.