His role as a racial pioneer should be emphasized. Including Sammy Davis Jr in the "Rat Pack" had important cultural implications. If the chatter on his albums is at all symptomatic, he had a keen appreciation of the roots of his music and its state-of-the-art practitioners.
I can't dispute the information on Sinatra's left-wing sympathies, and nobody here can read his mind, but the words of "The House I Live In (That's America To Me)" are explicit and not quite pure left. I would call them democratic/patriotic, which goes back to a basic reality I have noted in prior posts: the roots of popular, progressive insurgency are more populist than Marxian. By Marxian I mean the categorical rejection of the institutions of markets, capital, bourgeois democracy, religion, and nation-states, to name a few items. The Cultural Front was populist, by and large; ties to the CP are not pertinent in this context, since the CP's concessions to popular sentiments need no exegesis. Regardless of whether you think that's good or bad, it was real.
The full title of the song I note above may have been changed to include the words in parens in subsequent years, reflecting McCarthyism and all that, but the words are pretty explicit.
LP implies Sinatra's shift to the right mirrored that of the public, but this may be giving him too little credit. There is the story about the Kennedy Administration snubbing Sinatra, a former loyal supporter. It's possible that Sinatra would have upheld a basic working class value, loyalty, despite the shift of his audience, but for the disloyalty of JFK. His goombah Tony Bennett seems to have kept the faith (marched at Selma and sang 'Give Peace a Chance' on the eve of the Gulf War with Lenny Kravitz and a few others on the Arsenio Hall show).
Son of New Jersey,
MBS