Every shittah in this debate probably can be traced to a single source.
Certainly, anti-zionism is central to Satmar, but it’s far from their sole reason to exist. It’s a part of their vision (or version, for the cynics) of pure, unadulterated Yiddishkeit, but not the only part.
DY: I don’t think you understood my point. Chassidish Rebbes are much more likely to follow the shitta of the previous Rebbe because of who he was (and his status of Rebbe flowing from his predecessor). This is opposed to a Litvish Rov (e.g. Rav Chaim Shlita), whose status is not dependent on who his Father or FIL were (even though they were Gedolim). Therefore someone like Rav Chaim has the latitude to disagree with those before him, while a Chassidish Rebbe has much less room to disagree.