I just spent a long time organizing and sorting out the "stuff" on his son, William Smith Bryan's profile, and I did work some also on the next generation Francis (presently numbered III).
Sometimes progeny can help inform immediate ancestors, but also these references might have more than I had the patience to review:
"William Smith Bryan, transported to (Virginia), as a rebellious subject, settled in Gloucester County; Francis (d 1694) returned to Ireland in attempt to recover hereditary titles and estates, sought refuge in Denmark, where he m Sally Brinker;" citing Abridged Compendium of American Genealogy [ note that this Francis is the son of "WSB," the source was already on WSBs profile, just lost in the chaff]
One source that was used was sort of jumbled in its presentation, and I hope that that is cleared up now: Colonial Families of the United States of America. (Still, it feels like these profiles are "defensive" as opposed to objective. As if inundating the same statements over and over is going to make them come true. Some things that were stated were supported by the sources, but it was not apparent because of all the chatter.)
Lastly, for this Francis' son "WSB," another independent confirmation of his arrival in 1650 was found:
Apparent match on a migration record: US and Canada, Passenger and Immigration Lists Index, 1500s-1900s: William Smith Bryan, Arrival: Virginia 1650 Ancestry Record 7486 #3967835.
On Francis III's page, I found a source that is among the "early" ones (definitiely not the earliest, and nowhere "near event"): The Shearer-Akers family, which does make some small mention that WSB descended from: "Sir Francis Bryan, a prominent man in Ireland, about the middle of the sixteenth century, and probably President of Munster, (who) is believed to have been the ancestor of the following Bryan line:"
Gen. I, Bryan No. 1 is William Smith Bryan.
That's all I've got, right now.