TA and qualified first aider here!
This is what would have happened at my school, no ambulance as it is only required if bone is sticking out of skin.
Parents called to take child to hosp, if can't get through to any named contacts, messages left and member of staff takes child. If no safe way to travel (booster seat / insurance etc.) ambulance would have to be called and member of staff accompany. This would be a last resort.
Icepack - perfect. Helped to reduce pain and swelling.
Lots of TLC. Some juice and a 'be brave' sticker.
To me, sounds like the school did a great job but I understand as a parent why you're unsure and feel the way you do.
A first aider has literally only had an afternoon of training. They're not a doctor so wouldnt actually know for sure if its broken.
We dont have slings at our school. Just bandages. We were taught in our training that slings could do more damage if done incorrectly and it would be better to support the arm without if you weren't confident. We spent about 5 mins practicing slings a couple of years ago. So if this was the same for your school, this may be why they chose not to use a sling.
I redid my training last month and slings weren't even covered! It focused now on choking, CPR and poisens etc. The proper life saving stuff. We only went through applying a bandage and supporting arm using bandages if there's something stuck in the arm or leg or wherever
mind you, it was pediatric first aid so perhaps this is why. However both different trainers from different companies and im aware the guidelines chop and change regularly so maybe they are up to date or not up to date.
Hope your dd's arm mends quick!