Yes background checks could be bad, although if rejection had to be for specific cause, and could be challenged in court, not too bad. If rather than "background" checks, they were simple "eligibility checks" before being placed on the primary ballot I think they would pass muster. As it is, several candidates who where ineligible were removed from ballots in the not too distant past. Two, Eldridge Cleaver and later Larry Holmes, were not old enough, another, Roger Calero, was foreign born.
Simple majority elections do not override the Constitution.