It seems that no one who posted knows how to learn a Be’er Moshe (especially not the mechutzof who undertook to argue on him whithout reading what he says). The Debriciner is discussing a shul whose permanant Shliach Tzibbur has a television at home. He says that since the Shulchan Aruch (Rem”a) says that the halacha is that we warn, and then remove a Shliach Tzibbur who sings inapropriate songs, all the more so we must get rid of a shliach tzibbur who is known to have a TV in his home. He adds that one may not daven in such a shul if they don’t remove him. He is not discussing at all whether such a person may occasionally daven for the amud.
I agree with the OP that anyone with unfiltered internet is certainly no better than one who has a TV, and that such a person may not be retained by the tzibbur to be their official shatz. He also says that if a person’s aquaintances know it is considered common knowledge.