The Wolf:
1) There is no ”problem” with Rav Chaim’s position.
2) Rav Chaim’s position is in concurrence with ”common practice.”.
3) Rav Chaim is in fact following an open Mishna Berura.
4) ”Most of the world” takes this approach.
5) In fact, not one poster on this thread thus far has cited a specific identified opposing Rov, and certainly not anyone near the stature of Rav Chaim.
6) Even if someone of such stature is found, the disagreement at most will be whether minyan or hat/jacket takes precedance. There will be no disagreement with Rav Chaim by anyone of stature over the critical nature of a hat/jacket where possible.
The above may not suit well your American ears, but the Emes is the Emes.