Reeder: “Federal tax on dance floors”, indeed. But I do think you’re onto something with “they wanted all the room for couples”, as is Jomo Mojo with the wartime shortage of men.
A rule like this pairs women and men. It also sets up the club with more customers.
My thought is that just like in Jr High School, the women that want to dance would dance with each other if they couldn’t get men to… in wartime, of course this would be because men were in short supply. While it might look odd to some to see women dancing with women and possibly repel some of these customers, I don’t think that’s the main intent of the house rule.
Where the rule for the women is that “If you want to dance at all, you’ll dance only with whatever men are in here”, it means that it’s easier for men to break the ice with the women. This in turn draws more of the scarce male patrons for the club. Which in turn draws more women, and so on. The women like it, and the men like it. If you’re selling drinks and dinners, it’s good for business. So the club likes it, too.