You are so right. I've read pretty convincing arguments that conclude that Shakespeare's Richard III was based on the writings of John Morton, the Tudor's Joesph Goebbels.
It helped to have Shakespeare, the popular artist, propping up the myth back then. I just had the BBC "Elizabeth" documentary from the library, and the play Richard II is mentioned. When it was first printed (probably what we would consider nowadays to be a bootleg text) one of Richard's soliloquies was cut because it made reference by analogy (intentional or not) with a then-current power struggle between Elizabeth and one of the members of her inner circle. Later, Shakespeare quoted (in one of his plays, perhaps "The Tempest"?) the Latin poet Ovid -- "treason doth never prosper -- what's the reason? If it prosper, none dare call it treason." There couldn't be a more clear reference to Elizabeth's grandfather, Henry VII, but there's no way to know whether the playwrite intended that.