Hi Jamie; that particular piece of research is based on a collection of letters collated, primarily, by Colonel David Stewart of Garth and, although once published, as The Stewarts of the South, no longer appears in print. A transcript is noted off that site however. They are regarded as having some authority. Although we can assume considerable confusion on the individuals due to the Jacobite Rebellion, the Alexander Stewart, son of Glenbuckie, that page refers to, was younger than Duncan Stewart, his older brother. Duncan was born 5 Dec 1724, according to that page, but appears as Duncan Stewart of Glenbuckie, during a State Trial in 1752, where he is confirmed as 27 years old. Hence Alexander must have been born in or post 1725. That page suggests his younger brother, John, was born Dec 1725 and so it might be reasonable to assert they were twins.
From what you mention your Alexander seems to have been born 1722 which would place him born prior to the marriage of his father to Janet Drummond. It seems unlikely he was of this family. The Stewarts of the South does notice a couple of the line that migrated to America though; one an Angus and another, not named. The Angus is said to have had family "three sons' according to the document.