Ever since I was a fair coed in the 1960's, I have always thought it strange to sing, "I am a Utah man."
If the University of Utah had started out as a women's college, whose "fight song" began, "I am a Utah woman and I live across the green," you can be sure that, as soon as men were let in, the words would have changed instantly, because no man, then or now, wants to sing, "I am a Utah woman." It's just as absurd for a female to proclaim in song or otherwise, "I am a Utah man."
As for the other lyrics (all probably penned by a handful of frat boys over many beers back in pre-Title 9 days), if men can stand to call themselves jolly, I see no reason women shouldn't still be fair. But it's time to teach the new kids that they're "a Utah fan," not "a Utah man." And everyone who isn't a troglodyte can just start singing it that way at games.