Re: "…I am a subject to someone who was never elected…" —the problem is even worse than that. You may accept the authority of a surgeon who is not elected, but his authority may be based on something that you intellectually respect. If not, of course, you can try to replace him (i.e., find a better surgeon). The royal family fails even these (less-than-democratic) criteria for legitimacy: they don't study for and write exams that would establish their authority —and they can't be obviated if they fail to live up to our standards.
I Killed Queen Elizabeth II
Political Revolution in the 21st Century: How Not Why.