I can't see these FA rules lasting. They read very much like an extreme trans activity got in there quickly a few years ago (when people didn't really understand the issues) and persuaded the FA governing body who were not really paying attention that this was all ok. With all the light being shined on it now, it has to be reviewed.
I guess they just transplanted their approach wholesale from their handling of racism and homophobia. Given a long history of that in football, they're quite heavy-handed on those too.
But it's not simply analogous. They're not running separate gay and straight leagues, and separate white and black leagues. But they are running separate male and female leagues.
Even if there is real "transphobia", there is also the fundamental issue of people being permitted to compete in the wrong sex class. That's the debate here, and what they've got wrong. And as long as they're getting it wrong, opposition will keep increasing.