What the Mishnah B’Rurah actually says (91:12) is “And nowadays one must wear a hat while he Davens like the way he walks in the street, and not with the small hat that is worn under the hat because it is not nnormal (proper) to stand like that in front of important people.”
He explicitly states (twice) that this necessity to wear a hat is only time-based and because that’s how it was considered proper to present oneself.