Ok, I think the Batman analogy works well for US - it isn't important who's playing Batman, in fact it's irrelevant, as long as they want to do it and take the role on and commit to it.
We don't have any reverence as such to the actor playing Batman, but the role of Batman represents a place in our political system which is recognised as a "service" and "commitment" beyond any normal job - and we respect that and the person willing to honour that role. Full time life of army style commitment that doesn't end for the various people in the RF, but it's the Queen's show.
Harry isn't special or royal -he's a regular guy now, he doesn't have the mystique of someone who gives their life over to public life that none of us would want to do. No problem with that, but he isn't "royal" anymore in our eyes.
The person isn't royal, different or special - their dedication to the role is what is being respected.
It's the role, not the person so the US are wildly misunderstand anyone's importance as such - it is respecting the life of service they live - a bit like some feel honoured to meet the Pope - it's the position and service, not the person.
I think some in the US might still think the royal family are special in that they're better than us. We don't think that, I know some of the republicans think that is the case - it's respecting the RF commitment to service.
Not saying it's perfect, but that's what royalty are really - Batman franchise
I don't think in modern times we would accept someone on the throne like Andrew for example (who should be investigated properly) and he would be required to step aside asap (into police station for questioning).