I was delighted to find this thread as I have exactly the same questions as you, but about another John Burton/Ann Barnett son, Andrew (b 1791) who I believe is the younger brother of your William, and my husband's 4x great-grandfather.
Whilst it is very tempting to believe the information shared in trees online, it has also always seemed odd to me that "our" Andrew should have been a labourer if his grandfather was a vicar and a scion of a well-educated and wealthy landed family. I am therefore very interested in the theory that there is a link to Augustin Burton of Braunston.
However, my feeling is that there IS still a link between the families, though it may loop back a few generations and thus not directly involve the Rev Bartin Burton. If you look at the history of the (shall we say) "more illustrious" Burtons of Rutland, three recurring forenames are Bartin, Augustin and Andrew and it is this last one which originally made me think that there might be a link; the suggestion that there is a labourer named Augustin in Braunston only strengthens this feeling.
You might find the section on the Burtons in this magazine interesting if you have not already see it:
https://www.rutlandhistory.org/rutlandrecord/rr08.pdfFinally, I wonder whether you have DNA tested? My husband has so we may be able to find a common link.