And Miriam and the women of Israel in the wilderness.

Miriam and the women of Israel in the wilderness danced separately.

Of course. Behind a partition.

Simply speaking, on the Fifteenth of Av it was for the purpose of looking at the women’s beauty (after all, it was for marriage), but in ordinary dancing there is a prohibition against looking at women’s beauty (regarding married women there is a dispute whether it is Torah-level or rabbinic, and regarding unmarried women it is rabbinic; Beit Shmuel on the Shulchan Arukh), even before lustful thoughts of transgression.

What’s the question? It’s obvious that if he will not have lustful thoughts about the women’s beauty, and will not touch them affectionately, there is no prohibition; and even great Amoraim danced with women ["he carried the bride on his shoulders"].
And if he knows that he may touch her affectionately, or look at her for pleasure, then obviously it is forbidden. So what exactly is the question? Examine yourself and decide.
It seems to me that to do this and not stumble, a person would have to be either an exceptionally righteous person or a great wicked person [whose heart is so coarse in this area that nothing beyond the dancing itself would be stirred], and the fact that people practiced something does not mean it was proper or that one can infer from it.

Honorable Rabbi, regarding mixed dancing, in my opinion it is clear that there is a difference between the examples in the sources above that you brought and the relevance of the question nowadays. For example: my brother is secular, and when he excitedly wants to show me the party he attended last night, I force myself to watch one of the many videos he filmed. In them it is hard not to sense the frivolity of the dancers—some of them (so to speak) with fringes swaying from side to side, eyes closed, drink in hand and drugs in their head, while next to them stands some woman in her own world enjoying the moment in exactly the same way.
It is clear that if the crowd with the fringes (not just “so to speak”) chose to dance next to a woman, even if it were done without touching and without frivolity etc., still he would not do it, out of the judgment of someone who truly fears Heaven—out of an understanding that it is simply not fitting for a God-fearing person. Only when we reach the spiritual levels and the clean culture that existed then could we think about it, and then it would not contradict fear of Heaven. Today we live in a different reality, in my opinion.
My brother is always angry at me when I tell him that I don’t want to watch his videos, and in general videos with women dancing—but he does not understand that when his life story began with cigarettes in high school, "wasting seed" toward its end, cigarettes and trips with mixed company, and parties every day and drunkenness—in short, secular immodest living, which I doubt does not have an irreversible effect on the soul—and here I am, in my third decade, still fighting every day on the basic level of guarding the covenant, with a filtered device, and having never touched a woman—
of course he will never be able to understand me. He is immersed in the "swamp" and will turn the world upside down to justify his condition, because in my opinion he does not understand how serious the situation is. Discussing this question is legitimate, but in a forum of God-fearing people, mentally clean, and not tainted by secularity and frivolity as I described—and I don’t know how much point there is in it, since I don’t think a Torah scholar would ever think about such matters on the practical level, not because Jewish law forbids it, but because it is obviously not appropriate, especially nowadays. I don’t know how objective I am, and what I wrote is pretty scattered, but just a point for thought..

You are conflating the question of prohibition with the question of policy.