You misunderstand. This has absolutely nothing to do with the fans. Nobody actually cares if they end up at the game. This is about politics, lawsuits and PR. Mostly politics, because at some point owners are going to want money and tax breaks for that new stadium.I'd be very surprised, even in the south, if they couldn't find enough paying customers that have been vaxxed (or have counterfeited vaxx cards) to fill the stadiums. Sure, you might be turning away the die-hard long-time fans if they aren't vaxxed, but there will be someone there to take their place.
Also, the teams/NFL wouldn't give two shits if people are bringing fake vaxx cards to the game. It would be as quick as an ID check when you give your ticket at the gate. The teams wouldn't actually care about stopping unvaxxed people from attending, it would be done purely to check the box that they are "trying to be responsible".