I think that there is balance to be had between supporting school and competing.
Varsity cheers for football and basketball but I can tell you there are kids on my team who like cheering at games (and do it well), but primarily tried out to compete. If my program were sideline only, I think the tryout numbers would look drastically different.
I won't say that I think cheering for sports is sexist. However, I draw the line at certain things that I think are beyond cheering for sports and turning the corner into "football booster club" territory. Ex: There are programs out there requiring cheerleaders to purchase snack items to hand out in goodie bags for football players and spending more time making football crafts than actually working on looking good on the sideline. I don't find that to be a good use of my kid's time. I can't think of any kid I coach who would find that to be enjoyable. (Yes, those things are fine in moderation of course! I also have kids who will CHOOSE to do those things for their boyfriends and such, but I don't see myself ever requiring it.)