Here’s the official story (my summary):
הרב עמרם בלוי, יהודה ויספיש ואהרון קצנלנבוגן פרשו מאגודת ישראל בשנת תרצ”ה וקראו לארגון שלהם “חברת החיים”. הרב יוסף צבי דושינסקיא זצ”ל העיר להם שזה נשמע כאילו רק הם חיים, ולכן השם המקורי נעזב. כעבור כמה שנים הוחל לקוראם “נטורי קרתא” בגלל כותרת כרוז שהוציאו נגד כופר הישוב, וגם הם קבלו על עצמם שם חדש זה.
Sounds simple, but where and what else did Rabbi Dushinsky have to do with these characters?! And they themselves admit to being anti-establishment troublemakers, even before their official “organization” (and mostly ever since).
So, here’s what I think really happened (just a guess):
R’ Katzenellenbogen (Note: unlike Rabbi Amram Blau, who certainly knew how to learn to a large degree, even if he wasn’t a true sage, Reb Aaron Katzenellenbogen was no Torah scholar, and everyone knows that) was probably starting a conversation with Rabbi Dushinsky, trying to get support of some sort (like he did unsuccessfully with the Chazon Ish, Rabbi Z.R. Bengis and others). When Rabbi Yosef Tzvi Dushinsky, head of the Eidah Hachareidis (בית דין צדק לכל מקהלות האשכנזים) and a leader of Agudas Yisrael heard these wise guys were going to finally unabashedly declare themselves their own leaders, and rebel against the duly constituted Beis Din and Kehillah, etc., he made a simple remark opposing (or at least not supporting) the organization itself, stupid titles aside.
Rabbi Dushinsky humorously referenced a gemara (Kuddushin 34a: גברי בעי חיי נשי לא בעי חיי or Yoma 11b)., but the Talmudic reference completely sailed over R’ Katzenellenbogen’s head, and the rest is history.
Of course, by that logic, it’s not as though calling themselves Neturei Karta, and implying no one else qualify as ספריא ומתניניא\scholars and teachers (from Yerushalmi Chagiga 6a-6b) is any better! (And the second name makes even less sense in context.)
This was no loving guidance and counsel by Rabbi Dushinsky to an organization under his auspices or even tacit support!
(I am surely getting the relations of the official organizations [Eidah Hachareidis, Agudas Yisrael] at the time wrong here. I don’t have the time to untangle the web.)
By the way, what did the real rabbis think of “Kofer Hayishuv”? Sounds like a vaguely legitimate levy to me…