We are loooooooooooooooonnnnnggggg past the point of saving the country's political discourse. When it comes to this topic, the American left has no meaningful impact, so we end up using the terminology that's been well established for clarity. Nothing you said is wrong, but it makes conversations more intelligible in some contexts to refer to conservatives (right-liberals) and left-liberals separately when talking about cultural issues (since that's the only real divide between them, we agree).