And children are often carriers of a disease before it manifests itself in physical symptoms. There is the “serious flaw” in your argument.
But, and I have said this many times were talking reasonable precautions here.
I don’t expect other parents to be required to wash their kids cloths after a ball game, just don’t bring nuts into the school. The existing policies don’t require a health screening before a kid ges to school you just cant send them if they have running mucus or a condition like chicken pox.
The two policies are far more similar than the way you tried to use it..