Richard III is a fabulous play. But that is what it is.
Richard III, far from the only usurper in the period, is not a direct ancestor of the present Royal Family. But he is its distant uncle, since it is descended from his brother's daughter, Elizabeth of York, sister of the Princes in the Tower, wife of Henry VII, and also the source of the present Queen's descent from Muhammad.
Following the success of The Hollow Crown, and now this today, the BBC is pretty much duty bound to put on the three Henry VI plays and Richard III. With no suggestion, though, that they are historically accurate. They are not. By the way, at the time of his namesake's break with Rome, Henry VI's cause was fairly well-advanced there. Something worth revisiting, perhaps? As for his kinsman, he should of course have a Requiem Mass of the Use of York, and be buried in York Minister.
Although we would not want anyone to delve into the fact that the incorporation into English Statute Law of the Papal Bull legitimising John of Gaunt's Beaufort children with Katherine Swynford once he had finally married her (following the death of his second wife, who is also buried in Leicester) had specifically excluded them and their descendants from the succession to the Throne.
Just as well that John and Katherine's great-great-grandson, Henry Tudor, married their great-great-granddaughter, Elizabeth of York. Although, while that was his sole claim to the Throne, it was not hers.