Honestly OP. Please forget all these unnecessary posts on here. There are so many people on MN these days who seem to get a kick out of putting others down, they love nothing more than a race to the bottom just to show how 'sensitive ' you're being in your parenting. I know who I'd rather have as a mummy if I was a 6 year old little girl.
My daughter was exactly like your daughter at that age. In fact, much more sensitive, as you say this is unlike her. At 6, my daughter wouldn't have even gone to the hairdressers because she was so shy and nervous of strangers touching her.
So did I force her, tell her to get a grip, push away her worries like they were nothing just because I, as an adult, know there's nothing to worry about? No, I cut her hair myself at home where she was comfortable. I nurtured her and gave her so much love, reassured her over a time that suited her, filled her with so much confidence over the years that I now have a fabulous, confident, wonderful 18 year old who is getting ready to go off to Oxford uni (fingers crossed) in a few months. She also excels at a tough mixed sport that she would never have dared try as a younger child.
Unsurprisingly she is perfectly happy getting her hair cut by strangers now! She has a gentle confidence that I've never seen in anyone before. And this isn't because I dismissed her fears until she was resigned to not bother coming to me anymore, it's because I spent time nurturing and reassuring her until her confidence was so high the world couldn't knock it.
Keep being the parent you are, you sound fab.