An aristocracy would come about naturally, but I disagree with you on a few points. The wealth-through-birth issue is born of politcs. So long as you don't do things like corporate welfare or make politcs out of taxes (taxes would for the most part be non existant in this system), you can avoid those kinds of political games.
Yes there are bad kings in history. For everyone 1 bad king you can name, there 1,000 that don't get credit. It's a good system. In America, Kings are given a reputation of being power hungry inept slobs because of George III and the politics of the revolution. When someone is born into leadership and is raised knowing what is at stake and how to get things done effectively, you generally get far better results than the popularity contests of today.
And historically, Kings before Henry VIII, were nothing more than the head of the strongest family, who also was in charge of the army and maintaining the laws and traditions of the realm by virtue of his superior ability to gather resources. Traditions eventually gave these family's divine justification. According to all older writings of England and other medieval countries, the King was expected to "live off his own", that is to say, his own investments and farmlands, etc etc. They only taxed normally in cases of heavy emergency, which is why Prince John's taxing to free his brother from foreign hands was so demonized in history.