This is an argument about 'golf clubs' is used by men's rights activism and I have seen it used on MN before. Maybe even by the same poster with a new name.
This argument does not acknowledge the historical nature of 'men's clubs' and the business that was done there where female people were excluded which amounted to negative illegitimate discrimination. Those clubs were places of power for men to make business decisions and access networks. This resulted in women being excluded from those business opportunities and it meant that there was a power imbalance directly caused by these clubs excluding women.
Golf is a great example of this only so far as golf was used as a vehicle for these business opportunities to take place.
However, there is also the sports aspect of golf where female people cannot compete fairly with male people due to the differences coming from testosterone effects on the body. Meaning, that including a female in those groups for 'business opportunities' resulted in that woman being seen at a distinction disadvantage as she will take more shots and take longer to finish the course unless she is exceptional compared to the male golfers.
That again results in a power imbalance when considering it from an equality in business point of view.
Your points don't acknowledge the safeguarding issues associated with including any male adult in a female only space. Nor do your points acknowledge how some 'men's clubs' exploited female people for entertainment and even sexual activities.
There are legitimate and not legitimate reasons to discriminate against groups in society. Excluding women from what was being used as venues and events to conduct business was not legitimate because it negatively impacted on female people. Excluding women from a male only group such as the Men's Shed which is therapeutic as well as for socialising is legitimate.
Excluding female people from a private men's club so that those men can exploit female people for 'entertainment', are you here on MN arguing that this should be acceptable? If that private men's club was not a business venue, and had strict policies against the exploitation of female people and was only for social activities, then you could probably argue that there was legitimacy in the exclusion of female people.
Your points really seem to be regurgitating trope around 'the fallout from earlier Women's rights activism'? So much so that your points seem to be incoherent and just plain ill informed about legitimate and not legitimate discrimination and seem to support sexism and misogyny instead. Whether that was your intention or not.