gavra- If you only want to pick on the last 2 lines of a lengthy 3 paragraph post, I guess that’s you prerogative. I was just mentioning the proper attire required at a particular horse race (although the speed at which some shuls daven can be considered a horse race. But that’s a topic for a different thread).
I don’t know where you get the idea that a hat is a begged isha. Until JFK decided to appear at his inauguration without a hat, that was what everyone wore.
My guess is (only a guess) that if a hat is a begged isha, its only for those who decided to stop wearing a sheitel or otherwise cover their hair (another slippery slope and yet another topic for a completely different thread).