The Tudor claim is just weird. In the end you have to argue whether the Beauforts were legitimized or not by the fact that John of Gaunt married Katherine Swynford. But then that still bares the fact that they were barred from the throne.
and even besides all of this, Edward IV technically - on a technicality - won the throne through conquest as well, even if it was argued that his claim was through his descent from Edward III via two lines.
Three actually. Iirc his mother Cecily Neville was a granddaughter of John of Gaunt.