@MyAmpleSheep
I take your point that Schedule 3 services represent a tiny slice of the legal landscape in which sex-segregation is permitted for an important reason, with strictly equal treatment of the sexes otherwise being the default.
But it's not limited only to those things that cannot possibly exist in a mixed-sex form. The mixed-sex pond exists. And, whilst employers are compelled by regulation to provide multi-user single-sex toilets, service providers can offer mixed-sex facilities, albeit with a different infrastructure and inferior safety and hygiene outcomes.
Mixed-sex services are worse for women than for men, but we don't sue every time, because life is short, and consumer choice exists.
The rise of trans means that the repertoire of mixed-sex services has expanded to include services which are nominally single-sex but selectively allow in opposite-sex 'trans' people. This may be direct discrimination against the selectively-admitted sex, and/or indirect discrimination against the other sex if female, but in a fact-dependent way only, per Swift J. There's been no decision on direct discrimination, and the indirect discrimination findings in Darlington and LS v NHS don't seem to have put the CLC off one bit. It's going to be a constant litigation slog. And sub-optimal mixed-sex provision by providers trying to evade the problem altogether will continue to increase, at the expense of consumer choice.
So the problem is not enough single-sex provision, when there are important reasons (trauma, religion, safety, decency) to want it. So my idea was to use the existence of – let's call them single-gender services – as a hook. 'If you provide a single-gender service, you must also provide an equivalent single-sex service, sufficient to meet demand.' Obviously I'm hoping that the latter will be so popular that the former will wither away, saving the trouble of lawsuits.
Belief discrimination comes into it too. A service provider which provides single-gender, but not single-sex, services is catering to the moral proprieties of transgenderists, by giving them a quasi-Schedule 3-type service tailored to their beliefs. The up to 80% of the population who are non-believers are not receiving an equivalent service.
Can the Ponds declare themselves an explicitly transgenderist service, and tell non-believers to sod off elsewhere if they don't like it? Maybe, if they meet the rules that apply to religious organisations. But that means admitting that it is a religion.