Nope. When Goldwater ran in 1964 the The John Birch Society were avid supporters of his candidacy. There is NO QUESTION that the quote was a recognition by Goldwater of that support. If you believe otherwise you are wrong but you are free to make your case if you can. Just denying the premise is not good enough for me.
Yep, and that was his problem.
Phoenix at the time was full of Birchers. They supported him and he couldn't disavow them completely but they were loons. Goldwater had originally run for City Council on a clean government (not expressly conservative) platform. It was to stop the corruption and get the hookers off of Van Buren.
As Senator he had to step in and save the Alaska Mental Health Enabling Act (The Siberia Act) from the screwball right and he wanted no truck with the "Eisenhower is a Communist Agent" crowd. He needed to separate himself from the leadership of the JBS and their wacky notions without alienating the rank and file.
The Catholic Church in Arizona and also the LDS had strong JBS ties. Frank Brophy (Valley National Bank and Brophy Prep) was a Catholic Leader and a major Bircher. He was influential enough that Goldwater thought that he could derail Goldwater's presidential run. Goldwater did not want to be associated with his views. The extremism statement was an attempt to keep the membership after repudiating Welch and his nuttery. It also served to diffuse the attacks that Goldwater knew were coming on his starkly conservative views.