Could you perhaps pick a parent of your dd's friends, someone you think trustworthy and kind, and ask them if it'd be possible for you to go with your dd to theirs on a playdate? Or two?
You could easily fudge and suggest that you're a bit of a nervous-nelly, helicopter mum who is trying to get some practice in at being braver with dd's independence. Trust me, there's lots of us around who had no traumatic experiences but still feel this way.
I think once you 'out' yourself a bit, the other parents will be happy to help you. They will likely agree to provide (a bit of) extra supervision, send you the odd pic, outline some detail about what the play date will involve (who, where, what).
At least, as the most anxious mum on the block, I've had loads of help with this. There's some gentle ribbing, but my house is very popular for play dates because the parents are sure their kids will be well-monitored. Over time, with practice I have built confidence in the play date 'thing'. The balance between risk and benefit does sort itself out a bit once you start. In a way, having lots of play dates teaches your kid what the range of 'normal' is, so they can better identify any 'abnormal' activities or feelings (and get help early).
I'm a work in progress, of course. I have a dear friend with 3 dc, who regularly gets up and, laughing, turns my chair so I'm facing away from her kids. She lets them free range far more than I ever could, and does so on purpose. She sees it stresses me out, wanting to pull them back from the stairs or sharp edges (or from throttling each other). So she turns my chair!