Why do people say put a coat on so you don’t get pneumonia if pneumonia is spread from a pathogen

Because when the body is stressed: ie; illness, chilled, or other stress, it compromises the immune response.

I think they surmise that if you’re in the cold then it takes longer to get to your homeostatic balance and so might make your immune system more compromised letting, or making, you more susceptible to germs?????

While fever could be an over ramping of the immune response, the same immune response is damped under colder body temperatures. Lower body temperatures also decrease the blood flow through the organism via vasoconstriction.
Although pneumonia is spread via pathogen, these pathogens are continuously being in check by host's immune system. If the immunity is subdued, there's a greater chance to developing the infection.