I have some additional information on both the immigrant William and his grandsons, some of which I discovered and some of which comes from Bill Churchill.
William was married twice.His first wife was Mary, as evidenced by a deed of December 20, 1683.By his first wife he had two daughters, Mary and Susannah or perhaps Catherine Susannah.By his second wife he had daughters Priscilla and Elizabeth and son Armistead.The three are mentioned in his will as his daughters and son, and Susannah's "work" is mentioned with her name but not her identity.There is no evidence of other children named William or Judith.
Virginia had a census in 1787 in which it is estimated that between 95% and 98% of all white male tithables can be located in the county in which they were living.The census is supposed to list the name of the person charged with the tax and also white male tithables above 21.Columns record the number of white males above 16 and under 21, plus other information.
Second-generation Armistead's son John is listed as "Self" responsible for the tax and with one male between 16 and 21.On the next line a John is listed with the first John shown as responsible for the tax.The book "Some Prominent Virginia Families" by Louise Pecquet du Bellet says that John married Emma J. Nicholas and had one child, John.
Armistead's son Henry died of pleurisy December 24, 1760 at age 29.In the 1787 census an Ann Churchill is listed in Fauquier County, where John, Henry, and Armistead, Jr.had moved.She had no males between 16 and 21, but on the next line is a Charles for whom she is shown as liable for the tax.
There also was an inventory for a Henry taken on Jno 1796 in Fauquier County.There also was a Richard Churchill in Fauquier County by an 1800 tax list.
Thus, there does not appear to be any evidence that any of Armistead, Sr.'s sons had a son George, as John seemingly had only a son John and Henry probably was the father of a Charles and perhaps of a Henry and Richard, but not of a George. There was a documented George in the 1600's, quite possibly related to Samuel of Northumberland County. This George's marital status is unknown.
It would appear in the absence of documentation that a George born in Virginia in the 1700's is more likely to have belonged to the Samuel Churchill family (which might include descendants of brothers or cousins of the immigrant Samuel) than to the William Churchill family.