I'd assume in practice it would require a school to be accredited, because I think that's the kind of rule that the bureaucracy would instantly put into place, and that a president would never overrule because of the potential for bad press.and the fiddliness of it.
Accreditation isn't an enormous hurdle, but I've heard one or maybe two stories of schools shutting down because they couldn't make the bar.
The specific-degree requirement you suggest is probably for the best but may be politically a harder sell.
I imagine accreditation is a hard sell at this point too on the right given that the right may see a connection between government accreditation and the indoctrination they say is rampant in the higher education system, and the desire I imagine to create more religious schools that may benefit from not being accredited.
Accreditation isn't an enormous hurdle, but I've heard one or maybe two stories of schools shutting down because they couldn't make the bar.
The specific-degree requirement you suggest is probably for the best but may be politically a harder sell.