A noticable aspect of conspiracy-theories is how often the goals that the conspiracists were allegedly trying to attain were never reached.
For example, the OAS assassination attempts on De Gaulle were a CIA plot to stop De Gaulle from pulling France out of NATO(because, Lord knows, the OAS wasn't really upset about Algeria or anything). Then, five years later, De Gaulle actually does more-or-less pull France out of NATO, and the CIA doesn't bother doing anything about it.
Roswell and UFO's are very big yes.
What about "Pearl Harbor was deliberately let happen by FDR," "fluoride is a Communist plot" and "there's Red agents in the government and the medias"?
While UFOs were big before Kennedy, and there was always an element of "the government knows more than it's telling", the conspiracy side of it didn't really get going in a big way until after Kennedy's assassination. That didn't really happen until the late 70s/early 80s, with the Bennewitz Affair, Milton Cooper, alien abductions, and UFOlogists' discovery of the Roswell crash - although the Roswell incident made a few headlines in 1947, it had been completely forgotten until Stanton Friedman's interview with Jesse Marcel in 1978. In the early 60s we were still firmly in the lights in the sky/space brother period of UFOs. Whether that change would still happen would depend a lot on the general course of history with no JFK assassination.
Not that big. The John Birch Society wasn't exactly the mainstream, nor even more than a minority of a minority, problematic as that minority was in all the back alleys and basements of American society it found itself in.
Well, there's also the fact that canadiate Goldwater refused to renounce them, which probably gave people the impression that they had influence in high places.
Project U.F.O.
I think it was also known as Project Blue Book, or at least closely identified with that phrase. I took it pretty seriously as a kid, and along with Close Encounters, it is probably one of the main reasons that I, too, associate the late 70s with the heyday of UFOlogy.
An influence so great that it accounted for a landslide loss in the Presidential election, and a Democratic sweep of Congress.
The 1970s really was the heyday of UFOlogy, or at least of UFOlogy as UFOlogy. After they discovered Roswell and alien abductions - and FOIA - most of UFOlogy (not all) soured into just another branch of conspiracy theories.