If they could, what would be the point? They don't want to punish you; they want their money back. It's not like they could sell your degree to someone else.
I suppose just as an incentive to not default. If you keep the degree, in theory you are more employable. If you have to give it up to default on the debt, maybe you'd think twice?
If you implement it properly, there will be no transcripts or verification if bills are not paid.
I went to Harvard, its just I lost the degree in bankruptcy, so they claim I didn't go. Just call my references (which are actually burner phones held by my brothers)
Interestingly higher ed already requires all bills to be paid to graduate... they don't care if you default on your credit card the week after graduation, but you aren't getting the diploma until the library fees and tuition are paid off by "someone, somehow". Once its the loan servicers problem, they don't care and you get your diploma. I vaguely recall having to get a slip from some different offices proving I owe no money to the school, in order to graduate, and threats they wouldn't provide transcripts to students in collections, etc.
But your future employer may care about indications of your character and future behavior while under their employ. I've heard of companies running credit-history checks to discern whether a prospect is a risk of embezzlement.
I'm not saying the judgement / assumptions would be correct, but it's quite possible some employers would act this way all the same.
Therefore it's a big incentive to pay your money back.
But presumably if the employer cares about defaults it won't matter whether or not you get your degree since the default will show up on your credit record.