It doesn’t have to come out today tax payers pockets, that’s the entire thing. The money doesn’t exist, the debt doesn’t exist.
We made up this system specifically so we didn’t have to keep exact books, that’s the point of fiat currency over backed currency. If we don’t use its primary feature for good, ever, we may as well go back to the gold standard which would elimate nearly all banks and lenders at this point in the capitalist finite curve.
Except it’s new money that’s made up, and in the case of student loans, most of that money isn’t traded for goods and services, instead more than 3/4s of the money created goes back to the lender.