Even when I was in grade school in the '90s, the school's zero-tolerance drugs policy was that ANY medications, no matter what they were, had to be kept in the nurse's office in their original packaging, and administered by the school nurse as needed. Violations were punishable by suspension, and in theory at least this included things as innocuous as aspirin and ibuprofen tablets.

If you wanted to legally take any medications at school, you had to have a doctor's note and bring your own bottle (for prescriptions, only the original prescription bottle with the prescription label attached), leave it in the nurse's medicine safe (which only held student-owned medications; the school didn't provide even aspirin), and get permission from your teachers to go to the nurse's office to take it at the intervals specified in the doctor's note.

My memory of this is quite vivid, because in fifth grade (aged 10, in 1991) I got in trouble for taking Zinc tablets to suck on for a sore throat. I wasn't strip-searched or anything (fortunately Zinc tablets have a distinctive odor), but I was given a stern warning that the next time I was found with unauthorized medications I'd be suspended from school.