In a system of loans directly from the government, there is still some mild disincentive against lending to people that won't pay it back, even if it is tempered against some other incentives. What's more, a government body certainly can and surely would limit total loan amounts for affordability reasons.
More to the point, the comparison here is against regular private lenders, but instead against private lenders with government guarantees. Those have no disincentive whatsoever to avoid people who can't pay money back or to limit the loan amount. In fact they have a very strong positive incentive: there's a guaranteed profit on every loan and it's proportional to loan amount. What's more, if you cap the loan amount then the "customer" might use a competitor with a higher cap, so you lose the whole of the loan.
I believe private lenders require cosigners or else proof of income. I don't see how they can win here. You'd have them discriminate in giving loans based on some kind of lifetime prediction of someone's earnings? I'm sure that would go over well.
Schools need to be the adults in the room, since they getting most of the money after all. And that will only be motivated by having them take the financial hit if students default.
In fact if private lenders enjoyed the guarantees, i can imagine them sponsoring entire schools with full easy-button applications and admissions just to take advantage of the lending profits. Anything to push volume.
More to the point, the comparison here is against regular private lenders, but instead against private lenders with government guarantees. Those have no disincentive whatsoever to avoid people who can't pay money back or to limit the loan amount. In fact they have a very strong positive incentive: there's a guaranteed profit on every loan and it's proportional to loan amount. What's more, if you cap the loan amount then the "customer" might use a competitor with a higher cap, so you lose the whole of the loan.