OP, I haven't read the whole thread but I have been in your shoes. And how.
It is so upsetting seeing them hurt like that, isn't it? But she will be fine, honestly.
dd1 loathed sports day for the first three years. For her, the goal was never winning - it was just taking part, and one year she didn't manage that. On top of loathing sports and falling over etc (foot problems), she has stage fright and hates being watched doing something she knows she can't do well.
BUT. She is now in Y4, and shrugging about it (in two weeks time). She says she doesn't care really, its just a day. She likes running now, can't see why anyone would want to actually race, but whatever.
dd2 is now in Y1. She came second to last in her race last year, and has insoles and seriously inturning feet. She will never win. She sobbed afterwards, and asked why there wasn't a Paralympics (for flat feet!) at school. Her best friend is best at everything, including running. This year, she has decided, with my help, that running is fun and she will do it just to do her own personal best - never mind the others.
There was one event (a drama one) that dd2 came top of the whole school in: I used that to the max. 'Remember today next sports day, dd2. Today was your turn! Maybe sports day is someone else's turn.'
I could have written your post five years, and then one, year ago. But I promise you, you can turn this into a positive learning experience - if she is creative and good at other things, she will absolutely be ok. It's one subject, one day - not a life experience. Kids are mostly damaged by repetitive trauma, not one offs 