In 2008 I received an MSc in Scottish Ethnology from Edinburgh University for a thesis titled "A Study of Two Tunes: the Three Carles o' Buchanan and the Braes o' Balquhither in their Cultural Contexts from 1740 to the Present Day".
I come from Paisley, the birthplace of Robert Tannahill, and was always intrigued by the designation of The Three Carles... as the melody of his poem hence my research.

The argument is basically about oral transmission verses the printed page.

I reckon I can answer the points raised but very, very briefly here.

The McPeake's tune is entirely original and entirely different from the tune commonly sung by Alma Gluck et al although the the lyrics are obviously based on Tannahill's.

R A Smith printed in the Scotish (sic) Minstrel Vol 1 1821 the melody sung by Alma Gluck which very closely resembles the melody Burns used for And I'll kiss Ye Yet, Yet, Johnson's Musical Museum, 1788. Both melodies were variants of an old dance tune The Braes of Balquhider, John Walsh 1741, .....

I found an untitled sheet music dated 1810 for "The Braes o' Balquither.. Written by Robert Tannahill" and this is different from any of the above. Tannahill died in 1810 so his lyrics were set to music before R A Smith.
However Smith printed this tune under the title of The Three Carles ... In his Vol lV 1824 2nd edition.

It would seem that Smith's first version was copied by subsequent music publishers and Alma Gluck et al used this version.

The oral transmitted versions are much closer to the Three Carles ..... and Betsy Miller's, McColl's mother, is one such version.

Hamish Henderson wrote on the sleeve notes to John MacDonald's version that the tune was the Three Carles.... but in fact MacDonald sang the McPeake's tune.

Geordie McIntyre and Alison MacMorland sang their own tune as told to me by Geordie.

The Tannahill Weavers's tune is hybrid and doesn't cover the range of the Three Carles....

Greig/Duncan didn't pick up on the Irish songwriter Hugh McWilliams song The Lass Among the Heather whose lyrics became entangled with Tannahill's so much so that Jeannie Robertson recorded a verse from each song but called it the Braes o' Balquidder.

Hugh McWilliam's set his lyrics to Saint Helena which has great similarities to The Three Carles....

As I said this is a very, very, brief response to the points raised and my dissertation is too long to publish here.

I could loan someone a copy if required.