It seems that the actions of Richard and Henry are meant to model exactly what a good and bad ruler are like. Richard is many things- prideful, conniving, unwilling to listen to the advice of his uncles, and immature. But basically, he lives his life as though everything he does is right because God gives him the authority to do anything he wants. He doesn't feel that he owes his relatives or the lower subjects anything in particular. You can see this early in the play, when he feels like it's okay to take Gaunt's land instead of passing it to henry. We also see that he thinks it's all good in act four when he goes on about how God should save the king, if it's him, and the rest of them are traitors to God's will.
Henry, on the other hand, works hard to deserve the love of the common people and appears very loyal and dutiful. Even though he upsets the natural order of things, he does so to correct a perceived wrong. He is stoic, has an inward locus of control, and makes action happen.
If the two of them are a commentary on the way government should be done, then it would seem Shakespeare wants people in charge to feel a sense of obligation to the common people. The nobles should have some set of standards they must live up to, or they should risk being replaced. When he gives in to the pleading of Aumerele's mother, you can tell that he's a monarch who wants to please his people, at least to some degree.