“Just like policemen wear a uniform lehavdil so too us, we are the ambassadors of torah and we should wear a uniform including a hat. “
You can’t say that you are an ambassador of Torah while making up customs that are not in the Torah and passing them off as if they are.
If you want to tell me that this a social norm, then fine with me. But don’t judge others based on it. And don’t push them to have to conform to your social norms if they don’t want to. If, l’havdil, every kids was wearing a NY Jets shirt and yours didn’t, would you feel right about him being treated as less. It seems to me that many who wear a hat look at others who don’t as being less frum.