Can someone explain what the actual law is here though?
I thought that the EA states that gender reassignment is a protected characteristic. And 'gender reassignment means proposing to undergo, undergoing or having undergone a process to reassign your sex.' - so really, self-ID.
Therefore under the Equality Act, the default provision is to not limit trans people and therefore the policy, by default, is to allow trans people to use any space they like based on self-ID. This is the legal default - and therefore all the policies you mention are not illegal.
However, the EA has an exemption where organisations can legally exclude trans people from single-sex spaces - toilets, changing rooms etc. are commonly accepted examples.
The crux of it here is... orgs can choose whether to allow self-ID or not. There's no legal or illegal about it? It's down to the organisation to choose. It's not illegal to exclude trans people from single-sex space, it just seems most don't.
That's the law whether you agree or not?
Happily will hear other interpretations.