n0m, I would say that I know both Gur and Satmar quite well and that almost everything you said about them is incorrect.
Furthermore, all groups trace their lineage to the Baal Shem Tov. It’s just in Lubavich they have a longer chain of ben achar ben (with both Menachem Mendel’s being sons-in-law) than others. But then again, Rizhin has an even longer chain because they are descendents of the Maggid of Mezritch.
And you wrote: “If he had an issue, he spoke to the gadol directly and frankly.”
When was it exactly that he spoke to Rav Shach, because he certainly had issues with him?