I didn't misquote you.
You wrote: The sauna can lawfully offer special sessions for females only on the basis of sex and for women including trans women on the basis of gender.
We expand your conjunctive structure ('and') into two parts.
The sauna can lawfully offer special sessions for females only on the basis of sex
This is true.
And:
The sauna can lawfully offer special sessions for females only including trans women on the basis of gender.
This is not true, as we understand the law to be at this time.
Men’s clubs, expanded to include women. Whites only spaces expanded to include nonwhites.
Men's clubs that expand to include women have to include all women, not a subset of women. It's not lawful to have a club for men and (e.g.) tall women, or women who identify as men.
"Whites only" spaces: it's not clear whether you consider a 'space' to be a service or an association. But white-only associations have never been legal, as colour is excluded from the list of PC's for which single PC associations are legal. And there is no schedule that allows exceptions to the prohibition on discrimination by race in the provision of services. It's a silly example.
I don't think you understand the structure or operation of the Equality Act 2010 very well.
..It is by no means is an admission that the letter is correct and that the complainant would have won in court.
When an organization has staked quite so much time, energy and publicity on being "trans inclusive" as the GG's have, to fold like that is very much an admission that the letter is correct and the complainant would have won in court.