It's not that these billionaire owners are intentionally trying to suck ass, it's that they just don't truly care if the team wins or not. They're going to make their nut regardless. So, maybe winning would require hiring a no nonsense GM who cleans house and starts running the entire organization like a professional operation. But the billionaire owner doesn't want that because he's got a guy he's comfortable with who lets him sexually harass the help and do other fucked up shit (Dan Snyder). Or they have people around them that are more concerned with their overall balance sheet involving the stadium being used for other purposes, etc. (Blank).
Also, the cap only concerns the players. You can absolutely save money on everything else involving football operations from the salaries of the GM, HC, on down. It's why the Cardinals are always Fs across the board when it comes to stuff like food for the players, amenities, etc. Their shitbag owner is a cheapskate.