...more qualified for their jobs
It's always the same problem, who decides what the qualifications are? Age and natural born citizen don't seem like too big a burden, but even there some want to relax.
Also, education opportunities are not equal, as we all know. So, right from the start we're filtering out a big class of people. Once you say they need a lettered degree, you're saying more than half are unqualified, and that they should be predominately white males with money.
So, now the universities control who could possibly become President. Don't want a Socialist President? Don't give a Socialist a degree. It's much easier to filter them out at that level.