The plot thickens. This is the Missouri book
by Wm S Bryan in 1876 - except it isn't. It's a reprint by W W Elwang in 1935.
The original book only had a one-page preface, and it only had the one William Bryan, the Welsh one on page 132, who immigrated to Maryland in 1650. Then it skips over 2 or 3 generations and a State to another William Bryan, in NC, supposedly a descendant, who marries Sarah Bringer and has a daughter who marries Daniel Boone.
You can make your own mind up about the gap, but WSB didn't try to fill it. Nor did he claim to be a descendant himself.
But the 1935 reprint has a long intro supplied by Elwang. He discusses the original authors. This is where we get the whole story about William Smith Bryan the rebellious subject, descended from Brian Boru
Elwang says he got the later part from Bryan himself and the earlier part from "other authors". No connection is made with the Welsh Bryan, NC isn't mentioned, and nobody marries Daniel Boone, though the name Morgan does seem to feature in both families.
So the story in the "Bryan" book is basically the Shearer fable, as embroidered by other people. But Elwang must have known there was something wrong with it, from the way he skates round the difficulties.
Ironically, he starts his intro with a paragraph about pedigree-faking.