Reading this I wonder if she isn't overprotective as such, but more anxious about coming to the party with all the other mums. I have a terrible habit, which I regularly tell myself off for and stop myself doing, but if I am somewhere where I don't know anyone else and I am accompanying DD I tend to straighten her clothes or hair up alot out of nervousness on my part. Of course I let her go off and play etc, but I do still do this and I wonder if this mum is doing something similar. She may be very socially anxious herself and not want her daughter to play musical bumps or something because she may become highly anxious or even panicky if her daughter gets hurt and upset and she thinks everyone is watching the way she is dealing with it.
My behaviour is completely strange to me really because although I'm introverted by nature I'm not shy or backwards in coming forward. The fact that she sends her DD to nursery makes me think it is less about her need to keep DD close in an overprotective way generally but more to do with that specific situation.
So I think YABU to tell her to leave her DD alone, but it would probably help to be really nice and friendly to her.