"I've interpreted this to mean those who are supporters in name only"

That is a logical interpretation, but I don't think you will find any authority who would support it. First of all, if I remember correctly, Burns simply kept the phrase from the earlier, vehemently anti-Jacobite song, which perhaps was well-known, and was addressing dyed-in-the-wool Jacobites, not wannabes. Secondly, the phrase serves a rhetorical purpose, lingering on the idea of Jacobites, giving the listener time to get their brain in the proper gear, and rejects any possibility of 'moral equivalency' right off the bat. It has that sense of, 'Hey you! Yes, you! I'm talking to you' (or 'them').

Never heard JBN sung to the Sam Hall tune, or vice versa.