You’re right, he is absolutely a libertarian. But remember, a libertarian is a liberal. A social liberal.
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The Gipper would disagree -- "I believe the very heart and soul of conservatism is libertarianism".
That is why they exist, for liberalism, they claim to share conservative economics, but their passion is serving the leftist social agenda, that is why they fight conservatism rather than support it.
Libertarians are soldiers of the left.