Chacham Tzvi, R' Moshe Hagiz and their generation's struggle

In Shaalos uTeshuvos of Chacham Tzevi, the very first responsa as well as quite a few others answer shaalos asked by one R' Shlomo Ayllon, towards whom it seems that Chacham Tzvi has a lot of respect.

Yet, his contemporary R' Moshe Hagiz in his work against Sabbatens, this R' Shlomo Ayllon is one of the top three embodiments of evil, along with a notorious Sabbatean prophet Nechemia Hayun yim"sh and his partner Miguel Cardoso. R' Moshe in no uncertain words takes apart works of these people and declares them to be unequivocal heresy of the worst kind, the very anti-thesis of teachings of Ar"i ha"K and in general.

Chachaz Tzevi condemned Hayun by explicit request of R' Moshe Hagiz to examine his works, and despite one responsa in which Chacham Tzevi defends a scholar from London who delivered a discourse in which he subscribed to "Deistic" views for which he was scorned by some members of the congregation, Chacham Tzevi very actively opposed Sabbateans for the most part of his life, and certainly for the entire time when his mature works were written.

Available biographic sources bring an account of a protracted dispute between Ayllon and Chacham Tzevi, as a result of which the latter had to escape Amsterdam. Ayllon and Hayun were in this together and both partook in vindicating the holy R' Tzevi Ashkenazi, apparantly drawing quite a lot of support from the local congregation.

Later in life there seems to be a retraction on the part of Ayllon, so much so that he refused any contact with Hayun when he visited Amsterdam, and he admitted that he wronged Chacham Tzevi for bad reasons. But for all intents and purposed Shlomo Ayllon went down in the annals as a person who gave material and spiritual support to the worst Sabbateans of his days, and was a keyy figure in the tormenting of sages who actively opposed the Sabbatean heresy.

Why then is his name all over Chacham Tzevi's responsa ? And is there a pattern of questionable, shortsighted deeds by the congregation of Amsterdam that had a deep and long lasting effect on the history of European Jewry ?