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Rabbi Chaim of Brisk separated from “Agudas Yisroel” after the Katowice conference.

His reasoning is described in “Mikatowitz Ad Hei B’Iyar“, p. 56 (find more background over here):

Rabbi Chaim Soloveitchik of Brisk was invited by the German architects of Agudath Israel to their founding conferences in 1909 and 1912, but afterwards he withdrew his support from it. Family members relate that Rabbi Chaim gave the following analogy to explain his opposition to the Moetzes Gedolei Hatorah: In the old times, everyone had a candle in his house to give light. It was a small candle, but a candle nonetheless. And even if someone did not have a candle in his house, there was always a candle in his neighborhood that he could use. But then they built an electric power station to supply light to the entire city at once. Once the electricity was running, nobody kept candles in his house anymore, and if, G-d forbid, the power station stopped working, the entire city would be in the dark, with no source of light.

Rabbi Chaim in his wisdom foresaw that a worldwide Orthodox organization could be a good thing, but could also be a very bad thing. As long as every rabbi is independent, even if some rabbis err, there will always be some still on the right path. But when all rabbis subscribe to a single organization, if something goes wrong with that organization, all of its members go down with it. With eerie accuracy, Rabbi Chaim’s analogy foreshadowed events that took place many years after his passing, when the Agudah activists in 1947-49 led their followers into full-fledged participation in the Zionist enterprise, without the benefit of any ruling even from their own rabbinical council.

(I doubt the last, editorializing sentence is factual.)

And his point is that “if something goes wrong” is not an “if”, but rather a “when”! (Not to mention related arguments.)

The electric power plant metaphor was also used by the Chafetz Chaim to great effect, see what we once wrote here.

But this serves as an argument against any organization not mandated by the Torah (Beis Hamikdash, courts) such as the state! Indeed, Rabbi Chaim Brisker was “anarchic”, not just in his “lifestyle”, but also in his politics, as opposed to Rabbi Yoel Teitelbaum, who was fine with any Goyish state except Israel (endorsing democratic government in Vayoel Moshe!) and furthermore would be completely on board with the same kind of super-oppressive totalitarianism (beyond limited statism, which, itself, inevitably prevents Jews keeping mitzvos from free will), as long as it was under Mashiach’s management…

His son, Rabbi Yitzchak Ze’ev was similar. This is how I interpret the Satmar Rebbe’s following testimony.

“Tiferes Yoel” (part four, chapter 145), as quoted elsewhere:

But what can this mean? One logically cannot be more extreme than the rabbi who called the State of Israel satanic and all involved parties worse than idolaters. So, I suspect he means as I said.

Rav Kook related the following story one holiday evening in his sukkah. The incident took place in Jaffa, where Rav Kook served as chief rabbi from 1904 to 1914. One Shabbat day, a secular photographer came and disturbed the Sabbath peace in a religious neighborhood. In total disregard for the local religious sensibilities, he set up his tripod and camera in the middle of the street and began taking pictures.

This public desecration of the Sabbath deeply angered the local residents. One man who was particularly incensed by the photographer’s insensitivity took a pail of water and thoroughly soaked the Sabbath-desecrater. Naturally, the photographer was indignant. He was so confident in the justice of his cause that he registered a complaint against the water-douser — at the beit din (religious court) of the rabbi of Jaffa, Rav Kook.

Rav Kook told the photographer, “I see that you fail to understand the severity of desecrating the Sabbath in public, but you should realize that your action was a serious affront to the community. You entered a neighborhood of Sabbath-observers and offended them deeply.

“Of course, the correct course of action for the residents would have been to rebuke you verbally. Perhaps you would have understood the seriousness of your actions and stopped. Had that man consulted with me first, I would have advised him not to throw water on you.

“However, he didn’t ask, but reacted spontaneously. You should know that on occasion, such impulsive reactions are justified. When people disregard societal norms and cross accepted boundaries, regardless of the implications for others, it is often the spontaneous reaction that most effectively prevents future abuse.

“Such an occasion took place when the Israelites were in the desert and Pinchas responded, not accordingly to the normative Halachah, but as a zealot: “Kena’im pogim bo” (‘Zealots punish them’ — Num. 25:6-8; Sanhedrin 82a). If Pinchas had asked beforehand, he would have been instructed not to kill Zimri. But since his act was done sincerely and served to prevent future violations, his zealous deed was approved after the fact.”