You can just say "football team." The 'men's' is completely redundant. Other than it might emphasize that virtually all of women's scholarship sports are funded by the 'men's football team.'
OF COURSE you don't have to say "men's football team"! The football team has such a strong tradition, Tennessee Athletics pretty much IS the football team!
And it used to be the women's teams were so well esteemed, they had their own department and name. Now they are pushed back behind the men. You could say "Volunteers" and "Lady Vols" and people would know immediately who you were talking about. Now you say "Volunteers", and the assumption is never the women. That's not equality, that's a loss of value. The women are pushed back to the back of the line, rather than standing alongside the men in their own line. But the former Lady Vols who have been speaking out have expressed it much better than I. You should read their words.
I respectfully think you are wrong about the scholarships. Football pays for the bulk of athletics, but i believe it is donors that cover many of the women athletes. I don't have the numbers. we can both look into that, but I believe you are wrong.
The basketball team will temporarily be called "Lady Vols". Hart will put a stop to that as soon as he can get away with it, but donors are giving him a hard time right now.
Nike is getting calls because Hart said they were involved with the decision to do away with the name and logo. Nike claims it wasn't their decision, because adidas owns the logo (they don't - of course UT owns it). Does it sound like Nike will be making gear with the Lady Vols logo? Not to me! Already the cheerleaders at the women's games are wearing the T logo, and not the Lady Vols logo. Hart is just waiting out the complainers. It'll be gone. To me, this is very upsetting, disappointing, and I don't even see a reason for it, other than simple low regard for the women athletes and their fans.
The Lady Vols never hurt football. They used to co-exist just fine.