I wrote the stuff you just quoted from Craig Cockburn. You and/or Craig left out something else I pointed out at around the same time, that there was a "Braes of Bowhether" fiddle tune published by Bremner around 1750. So something related was floating around before Hamilton was born.

Burns has a few examples of material which must long predate him - "Parcel of Rogues" was first used as a tune name in the 1740s, though the tune itself predates 1700.

But these examples doesn't show that folksongs were "regularly" getting into the broadside press, rather the opposite. They're rarities. If it happened regularly there would be many comparable examples (like, say, the list of sailors' calls in The Complaynt of Scotland). If a song was significant in the culture of the time, it would get mentioned in letters, diaries, chronicles and fiction. (Or in lawsuits, as Roud says).