It would have saved everyone a lot of time if you'd admitted that to start with.
Right, let's go with entry level understanding of equality. First of all, achieving equality is far more complex than 'treating everyone the same' because that often leads to indirect discrimination.
Imagine a secondary school where you are allowed five minutes to get from one classroom to another. If you are late for your next lesson, you get a detention and this applies to all children to be equal. Unfortunately, 11 year old Brian is in a self-propelled wheelchair and his geography lesson is on the third floor, right after English on the ground floor, and the lift is on the opposite side of the building from both classrooms. The fastest he can do it, if the other children give way to him and give him priority for the lift is 13 minutes, so he's getting detention every week.
This is based on a real example. Is it fair? No. Is it legal? No. But the adults involved thought it was, because their understanding of fairness and equality had not advanced since they were five years old. Sorry, did that sound snide? It was meant to, don't worry.
In fact, this is actually an example of indirect discrimination against someone with a protected characteristic under the Equality Act 2010, because the rules (and the thoughtless timetabling) mean that Brian and other children like Brian are being put at disadvantage because they are disabled and cannot travel fast enough to meet the school rules.
Now, let's move on to the legality of specific services. There are nine different protected characteristics: age, race, sex, pregnancy and maternity, gender reassignment, religion or belief, disability, marriage or civil partnership, and sexual orientation. Sometimes providing equal access to services means providing services for people with only that protected characteristic.
First example: you may have noticed that sexual health clinics run men's-only sessions, women's-only sessions and gay/bisexual men's only sessions. This is legal under the Equality Act 2010, because the people who attend these clinics wouldn't necessarily attend an all access drop in clinic. If the NHS refused to provide specialised sessions, when they are aware that multiple sections of the population won't be comfortable with attending a general walk-in clinic service, that amounts to indirect exclusion of those groups from accessing healthcare.
For our second example, we'll return to leisure centres. I doubt you see it at David Lloyd, but at cheaper centres and council-owned centres, they will often have sessions specifically for people with learning disabilities and their carers. Presumably some would expect me to be up in arms that I have a monthly membership, and yet I cannot go swimming for that particular hour and a half in each week. Au contraire!
I am fully aware that I have the entire week to access the pool, and that for those with learning disabilities, it's the only time they can go or the only time their carers feel it's safe to take them. The leisure centre isn't discriminating against me, it's fulfilling its duties under the Equality Act by providing a ringfenced session for learning disabilities.
And now let's go back to David Lloyd. The question here is whether women are attending the mixed-sex sauna sessions in parity with men. I don't know that, but David Lloyd management will know, and they are making their financial decisions accordingly.
If they can see that their current sauna sessions are heavily dominated by male users and they think it's because women are unwilling to attend mixed-sessions , then it makes financial sense, and complies with the Equality Act, to offer female-only sessions to improve inclusion. If they don't, they may in fact be indirectly discriminating against women.