The common belief that getting cold causes illness is not directly supported by the science. Viruses cause illness, not temperature. However, the relationship between cold and sickness is not entirely without basis. Cold weather drives people indoors and closer together, increasing virus transmission. Some viruses survive and spread more easily in cold, dry air. Cold temperatures may affect the immune response in airways, potentially making it slightly easier for viruses to establish infection. The coat does not fight the cold virus, but there are legitimate reasons why the advice to stay warm has not been entirely wrong.
Daniel opens the episode confident he is right. Mom is confident she is right. The episode resolves the debate carefully and closes on a compromise that is also the correct science.

Daniel is nine. He asks the questions, refuses to accept easy answers, and picks every episode topic himself.

Mom does the digging - finding the sources, checking the claims, and translating them into something a nine-year-old will actually sit still for.
Viruses cause illness, not cold temperatures. Getting cold alone does not give you a cold.
Cold weather drives people indoors and closer together, which increases the transmission of viruses.
Some viruses survive and spread more effectively in cold, dry air than in warm, humid conditions.
Cold may affect immune responses in airways, but the coat fights weather, not the virus directly.
“I'm still not wearing the itchy one.”