DaasYochid:
I think you missed his point about a propeller beanie. Many people take the example listed by the Mishna Berurah about a hat being respectful at that time, and use it as proof that a hat is required during davening. If it’s the hat that’s the ikkur, not the respectful mode of dress, then yes, even a beanie should be fine. I’ve seen people walk into shul wearing a Yankees cap. I once asked one why, and he replied that the halachah requires a hat!
If the ikkur is to dress respectful, and the hat was merely an example given that place and time, then not wearing a hat should be perfectly fine.