In summary: you should wear a hat and jacket for davening. Period. If you don’t, it’s still halachically permissible, but you should.
We all know there’s going to be people out there making stupid arguments contrary to what all the rabbonim say, and trying to pasken based on their feelings. This thread is not bringing up anything new.
Some shuls do have a hat bin. Jackets are a little less practical to do that with for obvious sizing reasons.
DY, just curious (not arguing, honestly just curious), if you could fit your arm into the jacket sleeve with tefillin, would you still not leave one arm out?