The US Libertarian Party is really rather removed from libertarian parties of other natures or typical libertarian ideology. As far as I can tell the major difference between being US Libertarian or Republican is that Libertarians actively do not care if they're hated. Notoriously, Penn Gillette divorced himself from libertarianism because the Libertarian Party got too obnoxious for him.

Typically, Republicans trying to distance themselves from Republicanism pretend to be conservative independents. Bill O'Reilly has asserted such as has others on the FOX News cast. (Tucker Carlson, maybe? I wonder if it's a FOX News editorial guidelines to assert party neutrality, even as they were giving Trump softball interviews or letting him rant on the phone for 60+ minutes at a time.)