While President for Life Rockefeller did help, the whole "beat the ever loving shit out of the South until it's dead, then shoot it again just for good measure" did a lot to curb the rise of the OTL modern American conservative movement. While Conservatism in America is still alive and well, it's less religious and more focus on things like civil participation, individual liberties, less government intervention (economy, taxes, regulations), an embrace of capitalism, slow pragmatic change, and adherence to the institutions of law and order (not the dog whistle kind - actual enforcement of the law in regards to contracts and such and the establishment of police forces to keep the peace).
It's not quite Libertarian in nature, but it means more towards that side than say, the religious right.