No comment on Lineham - haven't listened, and have mixed feelings about his approach to this battle - but wow, the above is frighteningly naive.
I don't really care about sports one way or the other. If a sports division decides to have it single sex, then that's fine, but it doesn't really seem to be about fairness.
Erin, there may be outliers driven by whatever you're suggesting above (phobia? I'm honestly not sure), but of course it's about fairness in the overwhelming majority of cases. Without single-sex sport, women couldn't play safely in a significant number of sports or excel in any. Eg. there'd be no women in professional football, rugby, baseball, cricket... ad infinitum.
https://boysvswomen.com/#/
[Mixed-sex spaces] all exist already and have throughout history. So there's nothing inherently wrong with it. Feeling uncomfortable with it isn't really a valid reason to make it a human rights issue, because people are uncomfortable with lots of things, and they aren't pandered to about it.
https://www.independent.co.uk/life-style/women/sexual-assault-unisex-changing-rooms-sunday-times-women-risk-a8519086.html
Erin, the main reason isn't discomfort, but proven risks to women's safety (and legitimate anxiety resulting from that). Men, including trans-identifying males, commit 98% of sexual assaults, and have up to 150% greater punching power. Why do you think charities campaign for women to have single-sex spaces in developing countries?
Either you already know this and are being disingenuous... or you don't, in which case, for your own sake, please (as the awful, patronising cliché goes - forgive me for using it for the first time ever, it's just that it really does seem to apply here) "educate yourself".