Any Shul that objects to someone davening there that isn’t of their particular group, is not a Shul that anyone should daven in.
I davened in a Chassidishe steibel for a very long time. Nobody was ever turned away because they were not Chassidish or Yeshivish and everyone was welcome.
I’ve seen the same thing in every Shul I’ve ever davened in. It’s probably rare that a Shul would discourage people from Davening there.