I would imagine (no data to prove on me right now) that just like with covid, the majority of community transmission of any illness (at least the type we are speaking of here) will be done through those who are asymptomatic. Just like covid, one can carry the flu without ever having any flu like symptoms.
This is why the vast majority of medical professionals I have spoken to (well technically all) have said that temperature checks are not needed (or as they said "useless"). It is funny, when my wife attended the "return to school/play" meetings where they discussed how to bring kids back, they always had medical experts on the call. The medical experts always discouraged against the temp check for that very reason, while the administrators and politicians would pull for it. It's theater, and make them look good to have it, but does little to no benefit really.