This is really really insightful. And has actually been really enlightening to me.
I wasn't quite as bad as your daughter OP, I didn't have the confidence to directly criticise my parents like that, but I was permanently in a state of being embarrassed by them. They never really did anything specific, looking back they were just normal people, but everything they did was too loud, too much, too attention seeking etc to me. I hated spending time in public with them because I felt like everyone was looking at them and cringing for me.
It's only really now reading this post (I'm 42 by the way so it's been a long time until I have realised this!) that I have actually been able to see that it was me that was the problem. I was an incredibly anxious and self conscious teen, it was crippling at points, and I had almost zero self esteem. I actually now think I was projecting a fear of me being noticed or seen as 'cringe' onto my parents and finding everything they did cringey.
I have wondered, as an adult, why I'm never embarrassed or concerned about being in public with other people in the way that I was as a teen. I never think that other people (apart from my children!) are reflecting badly on me or anything. But actually I think this is because I am now much more confident and secure in myself, and don't see everyone else's behaviours as a reflection of my own.
Honestly, this is a bit of a revelation.
So OP, what I would try to do is work on her confidence and self esteem. She may appear outwardly confident but inside she is dying of self consciousness and terrified that people are looking at her. I know this isn't an overnight thing you can change obviously, but it will be so worth it for her.