That is untrue.
It's true that conservatives took credit for Reagan's first election: they nominated him when the GOP, following the Yacht Club's lead, had twice repudiated him previously -- nomination necessarily preceding the possibility of election.
And I'm not even mentioning the fact that Reagan's clear conservative principles, and his articulation of them, produced a landslide victory over the Democrats, despite the caterwauling of the liberal press (but I repeat myself).
But it is untrue, and a liberal canard, that the John Birch Society was the core of the conservative wing of the GOP. They were around -- but they were not the leadership of the Right as you say they were. Ronald Reagan and Barry Goldwater were that leadership.
No where did I state that the JBS was the "core" of the conservative wing of the GOP. The JBS was the "core" of the far-right. And Goldwater was its standard bearer and for a short time Reagan hitched his wagon to the train. Which is ironic since Goldwater turned out to be to the left of JFK after leaving politics.