Let's put the shoe on the other foot--if a Democrat who didn't have a four-year degree was running for the Presidential nomination, I'd have serious misgivings about casting my vote for them.
I have a master's degree. Considering that universities have been generous in awarding four-year degrees for several decades now, I expect my President to have obtained at least the same educational level as the Front Office Manager at the Candlewood Suites a few miles from my office.
But, I guess asking the person in charge of hitting the Doomsday Button to have a four-year degree in communications or whatever bullsh!t degree most athletes get is "elitist."
The fact that it takes very little effort to get a "bullsh!t degree in communications" kind of proves the point that there's no need to consider a college degree as a prerequisite.