shoring up Richmond's weak-but-for-his-marriage claim for the throne.

Oh, indeed. Since the Tudors were still on the throne in Shakespeare's time, it was completely in his best interest to make their rule seem as legitimate as possible.

And, yeah - Shakespeare messed with historical fact for dramatic purposes. But what I found amazing is how much fact there actually is in what was on stage. I definitely wouldn't have wanted to be part of that family.