Well, there are a couple of things that stood out to me: that your daughter doesn't like the performing involved, and that she needs to work on performing in groups in order to get more comfortable.
Is she unable to handle anything in public, such as speaking to her teacher or contributing to a group presentation in class, or is she simply not the type who enjoys being on stage alone? When I was growing up, I was fine with speaking in class, and being in a quartet or choir, and I enjoyed being in plays in school, but I did NOT like being the lead, or singing a solo. Never have, never will. A supporting role, yes. I had friends who absolutely shone when the spotlight was on them and tried out for (and won) every starring role. It takes both kinds!
So if your daughter is too anxious to do anything in public, maybe the lessons would be valuable. But if she is normally comfortable in typical social situations, and just doesn't want to be a performer, maybe it's time to rethink the lessons. There are plenty of music lessons that don't focus on recitals and performances. They still teach a valuable skill, just without the public performances.
Maybe you could ask your daughter if she would like to learn an instrument if there were no performances or recitals. Or ask her if she'd like to be in a choir with no solos required. Or ask her what she thinks her own strengths and weaknesses are.
And is your daughter also in a sport, and learning a foreign language currently? That can be a lot for an 11 year old. We usually required one extracurricular item at a time, whether it was a sport or a lesson, and we found that was less stressful on everyone.