And I promise i'm not being obtuse. I'd just like to know why the comfort of that 1 person is more important than the comfort of the other people in the room.
It isn't. And transactivists have done real number making people think it is. The protected characteristic of gender reassignment starts to apply as soon someone says they want to transition. If fact they don't even have to say it. You just have to think they are. If you discriminate against them on the assumption that they are trans, you are in breach.
However, it was entirely understood that there would be times when a transwoman should be excluded, with or without a GRC.
The key phrase is 'a proportionate means to a legitimate aim.'
In this case, the proportionate means was to offer them a third space. And the legitimate aim was to not piss off/intimidate/upset all your female members.
They could quite legitimately exclude this transwoman.
The tamtrumming and going to the press part is to convince other gyms to not abide by the equality act.
Women, all women, just need to understand that they do hold some power in this scenario. If they want to leave a gym in their droves on the basis that it has become mixed sex, avoidance of that is, in anyone's book, a legitimate aim for the gym to hold.