The OP asked us, "What do we think? and Would we allow our kids to do this?" I answered those questions but, if you want me to answer all of the above, here I go...
1) If a boy did this routine, would MY concerns be the same? Yes, young boys are exploited very much in the US.
2) Would there be any concerns at all? Yes.
3) Should boys not join a cheerleading squad because it is a girls activity or because their classmates or friends might harass them? Totally irrelevant to the topic. Your question is about gender typing and not about if we find it inappropriate because it is sexualized.
4) Should my CP stop doing cheerleading because someone might associate negative stereotypes with her or because she will be associated with other highly sexualized sports cheerleaders? If AS continues down the path of becoming more like NFL cheerleading with skimpier outfits and sexualized dancing "yes" my child will be removed. I can not answer as to what you will do.
You don't have to parade your child anywhere anymore, phones take care of that for you. Ideally, I would like to agree with you, realistically, I can't. The US has a huge child sex slavery issue and my friend works with rescued women and children. If you heard the stories of how these kids were taken into it and what they were required to do, you would probably rethink your stance. These kids are cheerleaders, ballet dancers, football players, little leaguers that 90% of the time still live under their parents roof. These kids pictures are found online, shown and bid on, these kids are often invited to a "new friends" house, raped by adults and then blackmailed that they will post the pictures, videos or just kill their family members. Unfortunately, the first thing I thought of when I saw this was, this would be some diabolical individuals dream child.