HenryVII was not only his mother's son, he also had a father...
His father was (afaIk) son of Catherine of Valois, and a half brother to HenryVI, who died without issue...
Everytime a monarch has no issue, the question of who is the first or the most suitable heir is a problematic one. Which is why HenryVIII went to such great and tragic length to secure one.
Margaret Beauford was a great granddaughter of John the Gaunt and thus had a claim to the throne, albeit not an overly strong one. And I presume at that time several people with equally weak claims simply fought for dominance. The Tudors simply had the best connections and the most money.
RichardIII made several tactical mistakes in making enemies at too many different fronts.