Thank you @Ereshkigalangcleg , I really don't think I am missing the point.
Please do read my posts again @IwantToRetire .
Still if you think you have worked out that the act doesn't allow this you need to start writing to the hundreds of employers, and service providers who use the Single Sex Exemption every day to exclude trans women
Hint - quote: A group counselling session is provided for female victims of sexual assault. The organisers do not allow transsexual people to attend as they judge that the clients who attend the group session are unlikely to do so if a male-to-female transsexual person was also there. This would be lawful.
There isn't such a thing as 'the Single Sex Exemption'. Service providers and employers are using two different EA single sex exceptions.
There are multiple single sex exceptions littered throughout the EA and they are all context dependent. Some of them have an additional gender reassignment exception that can be used in conjuction and some of them don't.
Those that don't are as much use as a chocolate teapot for as long as the GRA exists.
There is a single sex exception that can be used by service providers and public functions (NHS, prisons, other public facing government services) as you have outlined here - Schedule 3, part 7, paras 26-28.
https://www.legislation.gov.uk/ukpga/2010/15/schedule/3/part/7
Paras 26 and 27 say single sex / separate sex services are lawful. Para 28 says that anything done in relation to the lawful provision of single or separate sex services will not constitute gender reassignment discrimination. Without para 28 - a gender reassignment exception - there would be no legal way of guaranteeing single sex services because men with a GRC are legally female. In law, they share the protected characteristic of sex with women.
Employers do not use the same single sex exception. They use the occupational requirement exception in Schedule 9, part 1 of the EA. This is a widely used exception that is not limited to sex.
https://www.legislation.gov.uk/ukpga/2010/15/schedule/9/part/1/crossheading/general
If it was not for para 1(3)(a) - a gender reassignment exception - it would not be possible to exclude men who are legally female from female only job applications.
Another example is sports. This is dealt with under 'General exceptions' at section 195. The wording starts off similar to Sch. 3 but descends into the worst conflation of sex and gender anywhere in the Act:
(2) A person does not contravene section 29, 33, 34 or 35, so far as relating to gender reassignment, only by doing anything in relation to the participation of a transsexual person as a competitor in a gender-affected activity if it is necessary to do so to secure in relation to the activity—
(a)fair competition, or
(b)the safety of competitors.
(3) A gender-affected activity is a sport, game or other activity of a competitive nature in circumstances in which the physical strength, stamina or physique of average persons of one sex would put them at a disadvantage compared to average persons of the other sex as competitors in events involving the activity.
https://www.legislation.gov.uk/ukpga/2010/15/section/195
It's utter BS but this is where women's rights to female sports reside in the uk. Without the gender reassignment exception in (2) women's sports would have to include men who are legally female.
I don't need to start writing to the hundreds of employers, and service providers who use the various single sex exceptions in the EA to exclude men who say they are women from single sex facilities and services because I understand the law and can point to the bits of the EA that they are using that make their actions lawful.
However, there are other single sex contexts which are much less well protected in the EA. Possibly the most important of these is membership associations (girl guides, WI etc.) which are dealt with in Schedule 16 para 1.
https://www.legislation.gov.uk/ukpga/2010/15/schedule/16/crossheading/single-characteristic-associations
The EA allows a club to restrict membership to persons who share a protected characteristic, e.g. sex. Notably this is one circumstance where it does not need to be a proportionate means of achieving a legitimate aim. You can legally set up a women only club just because you want to. But there is no way of excluding men who are legally female because there is no specific gender reassignment exception.