I run a kids' camp.
I am the safeguarding lead on that camp.
Eveyr child fills in a medical form. that form specifically asks if we can administer Calpol etc if indicated.
No signature = no medicine.
If that young man was a leader on my camp, I would be horrified. he has broken the safeguarding procedure.
He would be suspended pending investigation. (normla for serious safeguarding breach)
If found to be at fault, he would be required to do further training before being allowed to work at the camp again.
All that is assuming he is the first aider, if not the first aider, then that is a SECOND major safeguarding breach.
This isn't hysteria, this is good practice. No, the OP didn't need to phone the police, but it is a serious breach of safeguarding.
And to the PP who side 'that's why no-one wants to run clubs' that's rubbish, I consider the safeguarding rules and a good framework to make sure I run my camp properly, and that both children and leaders are safe.