Gadolhadorah,
If the minhag in the alte heim when entering the court of a king was to REMOVE your hat or head covering, why is the mesorah to wear a hat when davening to show respect for hashem, melech malchei ha’malachim???
Do you think Judaism originated in Europe? The custom to remove headgear likely originated as a sign of respect/subservience – a knight or warrior removing his helm to show identity, respect, and vulnerability. In ancient Rome on the other hand, covering one’s head showed subservience, which noblemen didn’t like to do, but it happens to be exactly what we are intending to convey by covering our heads before Hashem.