It's the same reason Farage immediately backed away from mainstream politics as soon as Brexit happened. His entire career was about sitting on the sidelines and complaining about the EU because there was a lot of money to be made from Euroscepticism. Then we actually left the EU - which, in private, he definitely didn't want and didn't anticipate - so he resigned in a flash and started complaining about culture war stuff instead. The sooner we realise as a nation that the likes of Barton, Farage, Morgan, etc. are just grifting hardworking people by winding them up the sooner we can start fixing society and actually trying to make things better.
If Barton actually believed any of the shite he was writing about female pundits then it'd be worth debating him on it, but the truth is he knows his career in football is over so he's pivoted to GB News/InfoWars topics in a bid to make a name for himself over there. He did the same thing about 10 years ago, just on the other side of the political spectrum. He made all this noise about how he was reading books now, wearing suits and round glasses, came out in favour of same-sex marriage, all that stuff. That died a death on Question Time when he exposed himself as the same thick bastarding troglodyte he's always been so he tried to get back into football.
He made a decent go of lower-league management and probably could have made a nice career for himself if he'd stuck it out, but it's clear these days that the real money - a regular stream of income that doesn't come with the pressure of football management - is just raging against anything perceived as "woke" and becoming a rent-a-gob representative of curtain twitchers and football hooligans everywhere. You can see it in his tweets that he's just seeing how far he can go before he gets "cancelled" (hence the Fred & Rose West tweet) and then he can spin that into a book deal, TV spots, etc. etc. He's already been on Piers Morgan's TalkTV show, hasn't he?
When he's got as much attention as he can from female pundits he'll move on to something else. "Women's football shouldn't be on the BBC" or "Gay footballers should stay in the closet" will probably be next. Then he'll see if he can get away with suggesting that "There aren't enough English lads coming through at Premier League academies" and if that goes down okay he might test a quick "There's too many black players in the England team" line. He won't believe any of it, he's just another Clarkson type who gets paid to pretend he's angry about something.
There's a whole industry for it and social media has given them all too much oxygen.