I don't think the royals are really more inbred than the average rural Europeans, who historically often were born, lived, married and died in the same village, and were related to everyone else there. At least the royals would get some gene diversity when they would marry people from other corners of Europe who often had at least some local blood in them.
Carlos is the poster boy for inbreeding in the pursuit of political power through dynastic rule and consanguine diplomatic intermarriage.
I do not know that issues of George III of the UK were hereditary. Other members of the English throne have shown mental instability, much of it probably inherited. But, as Miisa notes, the mental instability of non-aristocratic citizens was often hereditary, as well.
I don't think that hereditary madness is created by access to absolute power, but the presumption of absolute power to those who come to that power by means of heredity opens it to the vicissitudes of hereditary madness and eventual collapse in a dynastic line and the mad in control of 'absolute power'. I would posit an alternative allocation of such power by demonstrated merit and trust of the community, rather than familial succession.