Fox News is probably more responsible than anyone for the GOP going far to right.
But if you wanna blame anyone its Nixon and Gingrich, Gingrich for the hyper-partisan attitude , Nixon for creating a Suburbs+Interior West + Sunbelt Coalition which allowed the GOP to win without the North East and Urban Areas and once that happened it was inevitable for the GOP to move away from Rockefellerism towards Reaganism.
Heres my analysis of this:
Reagan type Republicans taking over the party was inevitable as the Reason Rockefeller Republicans dominated the GOP in the first place from 1940-1976 was that in that period the North East, and Urban Areas was basically the Geographical Area that dominated politics in the country in that period(due to the South being a one-party state, and the Suburbs being not that influential ) .That's the reason why the GOP was so moderate because they had to be to win areas they needed to win elections.
The 1968 Election basically showed that the GOP no longer needed the North East or Urban Areas to win elections, as the Suburbs + Sunbelt coalition was more than enough to give the GOP a majority in the electoral college and by the 1970s that coalition was on the ascendant which meant the GOP no longer had to rely on picking off areas and voters from the old New Deal Coalition in the North East and Urban Areas to win elections. The Sunbelt and the Suburbs were also significantly more conservative than the North East and Urban Areas and since that was the coalition on the ascendant it was inevitable that the GOP was going to become significantly more conservative.
Lastly in 1979 the vast majority of the GOP establishment supported Reagan in the primaries and not HW (Reagan had 90% of the endorsements which were given out in 1979:
https://fivethirtyeight.com/features/ted-cruz-is-just-like-reagan-in-1980-expect-people-actually-liked-reagan/)