All the relevant evidence should be in my previous post; I included it as a guide to others who might be faced with a similar situation.
Returning to this point, I am still bothered that there may be a danger that this could lead others in similar circumstances to draw conclusions from distant matches that cannot safely be reached. I do in fact have a very similar situation in my family where one of my 7G grandparents is alleged to have founded a dynasty in America following an emigration in the early 1700s.
I say 'alleged' because there is considerable uncertainty as to who actually went to America and if they had any descendants there. I have no distant matches at all in this line, but my sister has distant matches with descendants in both the documented English branch and the alleged American branch, as do some closer documented cousins from the English branch. So is this proof that the American family is descended from my 7G grandfather?
I would say not. Two things stand out in reaching this conclusion - firstly the highly dubious nature of many of the trees alleging the American connection, and secondly the sheer number of people claiming a descent from my ancestor. Rather than there being 'safety in numbers', the more Ancestry members claiming a descent from a particular person or couple, the higher the chance of one of those descendants appearing by chance among our distant DNA matches.
In fact, when I search the matches of my husband and his brother, I find they have a greater number of distant matches to supposed American descendants of my 7G grandfather than my sister does.
Weighing all of this up, I have to conclude that despite the apparent DNA evidence, it cannot
confirm either my research or the alleged American link.