To be precise Sarah L states the following:

This is part of what Stonewall want to achieve:

A review of the Equality Act 2010 to include ‘gender identity’ rather than ‘gender reassignment’ as a protected characteristic and to remove exemptions, such as access to single-sex spaces.

So it will be impossible to apply exemptions, because gender identity will allow anyone who says they are a woman to become one legally. And there will be no single sex spaces

Sarah L seems curiously reticent to talk about in what context that recommendation was made. It is, of course, part of the testimony, evidence, and advice gathered by the Women and Equalities Committee in 2015. Stonewall was one of a great many organisations to give evidence in a similar vein. Many notable women’s groups and organisations, including DV organisations, also gave very similar evidence; pointing out that they didn’t use the exemptions because they didn’t need to — there had been no situation arising that required the use of the exemptions. Names now associated with organisations like Fairplay for Women also supplied evidence to the Committee

And having listened to the expert testimony and evidence in front of it the Women and Equalities Committee made the recommendations that the PC of ‘Sex’ and ‘Gender Reassignment’ be replaced with a single PC of ‘gender identity’ that would function much better to stop discrimination

Sarah also advances the rather strange claim that having a PC of ‘gender identity’ would mean the end to separate services for men and women because it “will allow anyone who says they are a woman to become one legally”. Yet for almost all purposes of UK law somebody has the lawful right to use a single sex service of their expressed gender identity. A service provider is, of course, fully free to use their own judgement about the sincerity of that expression and that would be true regardless of whether the PC is ‘gender identity’ or ‘gender reassignment’ and ‘sex’

This is the reality of now. Trans women use women’s services perfectly lawfully because, of course, they are women and are protected from discrimination in the use of those services and facilities by EqA. Trans men use men’s services perfectly lawfully because, of course, they are men and are protected from discrimination in the use of those services and facilities by EqA

The use of these services and facilities by trans people does not change the nature of these services. They remain for all purposes of the law single or separate sex services. That is the reality of the now. It would not change by allowing trans people the simple dignity of having the state recognise our gender identity for the purposes of marriage and taxes