This is what Newman Hall has said about IsaacIII. "One interesting aspect of this problem is the manner in which the tradition is reported by Walter S Allerton.He indicates that he obtained his information from a cousin, Mead Allerton, who in turn obtained it from another very elderly cousin, Sarah Allerton, who married a George James and who died in 1858.This Sarah was the grand-daughter of John Allerton of Norwich, CT whom she never knew personally.The second hand information that WSA reports is that Sarah stated that her great-grandfather was Isaac the grandson of Isaac of the Mayflower.This is the sole basis for the whole tradition.No other basis has ever been proposed or reported."
Newman Hall questions Sarah A's memory, but I don't.I like to think that it was perfect.She would not have known herself whether she was was talking about Isaac III, son of Isaac II, or Isaac, son of Bartholomew.The name's the same!
There were also other English Allerton families, notably the enterprising and economically successful Staffordshire potters.These were not related to IsaacI or even to Bartholomew, but sometimes expediently claimed descent when establishing themselves in America.An example of such a spurious claim is William Allerton, born Birmingham, England in 1801, and listed in Walter S Allerton's book.
You and I are amateurs, and it's great for you to guess that you're descended from Isaac III, and for me to guess that I'm descended from Bartholomew. But we must carry on searching for the evidence.Let's try not to put wishful thinking on permanent record on the Internet when it might mislead the researchers who come after us.