@puppymaddnessI feel like you need bullet points to help you understand, rather than paragraphs.
The law states, for the purposes of The Equality Act (and maybe some other laws), that TW are male/men and TM are female/women (this is what is being referred to when sex is mentioned)
Mixed sex groups - allowed - no discrimination
Single sex groups - allowed - legal discrimination using the exemptions in the EqA
Having a group but only allowing some of one of the sexes - illegal - sex discrimination
Calling an organisation "women's" but allowing male people - not allowed
(Groups with fewer than 25 members have fewer restrictions)
To address your anti democracy point, I think there have been You Gov polls which show that the majority believe that TW shouldn't be allowed to access sports etc meant for women, which would appear to show that all the above is in line with the general public i.e. democratic