TR's fellow Mugwumps -- the Republicans who couldn't stomach Blaine in 1884 -- were described in sexually ambiguous terms, by red-blooded Republican stalwarts: Republicans said that Mugwumps had their mugs on one side of the fence and their wumps on the other. The Blaine Republicans called them "Assistant Democrats" and said that they "had their hair parted in the middle, banged in front, neither male nor female." The reformers took the name "Mugwumps" with pride.
But no one would have called TR a "sissy" to his face in his time. He was called a "dude," but that wasn't the same thing. In the West, a dude was an Eastern wannabe cowboy. In New York, a dude apparently was a wealthy, somewhat young, inexperienced and immature clothes horse. But I don't think the sexual connotations were as strong as Vidal makes them out to be.
As for Alice and Longworth and Borah, if anyone remembered them before hearing the story of their sexual adventures it's hard to think of them in the same way or accord them the same respect afterwards. It's not something one really wants to know. Not after 8 years of Clinton.
There is a funny story associated with Longworth, though, which you've probably heard. A gentleman apparently inadvertently touched Longworth's bald head, felt it and said, "Why, it feels like my wife's bare bottom!" Longworth put his hands on his head for a minute and exclaimed, "Why, so it does!" Whether the story's true or not, I don't know.
Lol!