Mine's a bit like this too, now 7. It has got better over the years. I totally understand you being embarrassed, because people who don't understand think they're being rude or standoffish. The number of times people have reacted as though their dc is being snubbed. When the reality is mine is like this with everyone. It takes her ages to warm up in social situations. She often doesn't respond to people she knows well who say hello in the street. Every morning the TA says good morning as she walks in, but she never replies. It's mortifying.
She's now of an age where I can try to explain to her - how do you think they felt that they came up and said hello and you didn't reply? And that actually it's far easier to just very quickly say hello rather than hide.
Going forward - mine is fine with one or two close friends. She can't really cope with more than that. So we have small parties that she can cope with, usually about people. If she's trying something new, I try to get a friend to go with her. She quite enjoys trying new things if it's a small group - so I try to find things like that.
As for school plays and reading out loud - mine still doesn't do that. But I just think there's no rush. I was the same at her age and I was forced to take main parts in plays which caused me huge amounts of stress. I'm glad they don't seem to be doing that to her.
I am now an adult and have done presentations and things for uni and work. I still have social problems - I don't do well in groups. But mostly I'm fine. She's just inherited some social anxiety I think. I think my aim is to try and manage it.
i don't know what the answer is. just to accept her as she is and try and increase her confidence slowly. it's funny but as time goes on they all shine in different ways. whereas some of her friends are the all singing and dancing types good at so many things and so confident, she has had prizes for artwork, maths, being kind. have faith in her abilities in other ways.