Well I think at this point you're just using puritanism as short-hand for "people trying to enforce morals I disagree with".

So if you're left-wing, than in your terminology puritans will be by definition right-wingers. And the people enforcing morals you agree with cannot be called Puritan.

If you take a brief look at the history of the original Puritanism however, it was definitely a 'revolutionary' movement against tradition. Including the ones who came to America.