In practice, board bylaws and common sense mean that individuals recuse themselves as needed and don't do stupid shit.
If OpenAI didn't have adequate safeguards, either through negligence or becauase it was in fact being run deliberately as a fraudulent charity, that's a particular failure of OpenAI, not a “well, 501c3’s inherently don't have safeguard” thing.
Most people just tend to go about it more intelligently than Trump but "charitable" or "non-profit" doesn't mean the organization exists to enrich the commons rather than the moneyed interests it represents.