The tune for StB is called "Brave Scotland" in the Gesto Collection of the 1880s (big Scottish fiddle tune book). I've seen in a manuscript book of pipe band tunes used by the Boys Brigade in Edinburgh during WW1, called "Scotland the Brave!!" - no idea where they got it. Maybe Cliff Hanley was in the BB? At any rate, there's no reason to suppose the route of transmission went through Ireland.

FWIW, Hamish Henderson's "John Maclean March" used the tune a year before Hanley wrote StB. Henderson just called it "traditional" without naming it when he first published the song.

It had never occurred to me that "Hail to the Chief" and StB had anything to do with each other.

Who was Sanderson? The Scottish broadside publishing family of that name didn't get started until about 1830.