I think I've come up with the simplest way of fixing the circular relationships. The key to it was Annie Middleton. She was George's wife and circuitously, his grandmother. I clipped Annie as a mother of Francis and now all is well... at least, logically if not factually. Here's what we have:
Thomas Besleg-1 1216-1272 (no wife), father of
Richard Beesley-16 (no date, no mother, no wife), father of
Francis Beesley-15 (no date, no mother, no wife), father of
Francis Beasley-265 (1550-1609, no mother, wife Jane UNK), father of
George Beasley-264 (1586-1635, mother Jane UNK, wife 1 Ellen UNK, wife 2 Annie Middleton m 1619)
Since the last two generations have dates that work out, we can at least say it is logical if not supported by evidence.
The three earlier generations lack dates, evidence, and mathematical sense with more than 300 years from Thomas Besleg to his great-grandson.