It should also be addressed that it is common among chabad to believe themselves to be at present and historically much more significant than they were and are in population and influence. Not to say that they were not significant before the war, but they were dwarfed by the enormous numbers of kotzk-school polish chasidim; Aleksander, Ger, and others were far more numerous, as were belz, klosenberg etc and Hungarian chasidim such as munkatch and sihgit. Even in russia, breslov was active, and the largest number were in Chernobyl-school chasidim, Chabad was and is not part of agudas yisroel, which made up the vast overwhelming majority of non-hungarian charedi jewry in all of Europe. Chabad made up perhaps 8% of charedi jewry as a liberal estimate.
They also link every rabbinic family – especially brisk, for some reason, to chabad, though in my experience they often conflate rav chaim with the brisker rov, and assume that rabbi yoshe ber has the same status in the yeshiva world as the former. In my yeshiva, whenever the missionaries would show up on yat kislev to share their “chasidis” with us (we weren’t interested and we’d often show them exactly where the doors were and threaten to call the police for trespassing), they would often talk of the one or two rebbeim who were chabad in the many years of the yeshiva; the whole world revolved around chabad – only their Judaism mattered and only their constituents were full “fulfilled” or “whole” Jews (sound familiar? Jews for J anyone?).
Rav chaim volozhiner was not “close” with chabad, or any other chasidim. He worked together with the baal hatanya briefly on klal matters. His sefer nefesh hachaim shows that he was not exposed to mainstream chasidic thought, due of course to no fault of his own (many achronim were critical of kabalah, because they were not exposed to it in full). Take for instance shaar 4 in the beginning where he says that chasidim think that Torah lishma means to think about Hashem the whole time – I’m sure there were chasidim who spoke like that, but that is not the way the rebbes actually held.
To say they were close is completely false. To say that rav Moshe feinstein and rav hutner were “close” with the lubavitcher rebbe is also false. There is no end to the amount of attempted association that goes on in chabad; they think that literally everyone in the Torah world is just shy of being a chabad chossid – everyone has some relative, talmid, rebbe or neighbor’s son’s mechutin who was chabad.