Wren, I don't think 'earlier exposure' to some kind of performance is the answer here. I don't think it would have made any difference for my D. Older D is a true extrovert, so we were continually surprised (and, like the OP, sometimes embarrassed) by younger D's shyness in various situations. I recall reading that it is possible that there are some genetic variations that increase the odds that someone will be shy -- that isn't going to be changed by making them perform in some way.
Here is a story about my shy D. In 1st grade her teacher was talking about Abraham Lincoln (coming up on President's Day), and D eagerly offered to recite the Gettysburg Address at the post-lunch "sharing" time for the whole school that they have every week. I don't know WHAT she was thinking, really. I mean, she did know the GB Address by heart, but if she had been considering this, she would have known that it also meant speaking in front of 200 people! So the day came, and I slipped into the lunchroom to watch her. She got up there -- and froze. Couldn't utter a word. The lower school principal came over to help her. They got a chair for her to stand on, and the principal put her arm around her. They started the first line into the microphone together, then D finished the rest by herself (to loud applause at the end, especially from the amazed teaching staff). So it ended up being a success, but it was also quite traumatizing for D. Until last year and debate (which has a very small crowd

), she has avoided public speaking as much as possible.