My Rosh Yeshivah used to make any members of the yeshivah who didn’t have their hat and jacket for whatever reason- daven in the ezras nashim.
We can easily reconcile what Rav Chaim said with common practice among many rabbanim.
The rabbanim of a shul have to weigh that:
1. If they send this person home- he may not daven at all.
2. If the person doesn’t wear a hat and jacket in general it may distance him even further from yiddishkeit by demanding something of him that he doesn’t want to relate to.
3. If the person is not a mekabel then it will just cause an argument with no real gains.
Also, if the rav doesn’t stop every single time someone starts talking in the back of the shul this would follow along the same lines, ie. he just doesn’t stop people from doing the wrong thing all the time.
Also, would the rav of that shul kick out someone who came to shul without tzitzis on? This is also no different- the rav is happy that a person at least comes to shul.