Real immigration reform like this requires legislative action which was taken of the table by house republicans because it is their strongest issue to run on. Trump is a symptom of this disconnect in government, definitely not a solution to 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.