I had kids giving themselves shots. 70’s and eighties. I remember a hemophiliac with a kit of bandages and shots he carried around.
Just give kids responsibility. They will do fine.
I think you’re focusing on the wrong thing. The problem is not that kids can’t use sunscreen effectively. Whether they can or not is irrelevant. The school has a stupid rule. The mother was too dumb to protect her own kid in light of it. Whose fault is that?