I think most people in the USA have only the vaguest idea of what royalty even means. That's why you see a lot of rappers, etc. claiming that they're kings and queens, because they have this idea in their head that it's only a status thing and that they can get people to serve them or see them as better if they claim that.
In the USA we are supposed to recognize that there's nothing magical about somebody's bloodline and that each person needs to be evaluated on their own merits without respect to who they are related to. Having a father that's a competent president for example, should not automatically mean that you would make a good president too.
In countries that insist on inheritance by blood, I think there is something lingering of a belief that the blood carries some kind of Divine or supernatural linkage which can only be passed down a family line.
I think there can be generational blessings and cursings but in reality we are all individuals with free will and our own talents and things we're not so good at. Really I think in the modern world countries shouldn't have a concept like royalty, but I'm sure that such systems will persist for a very long time regardless.
Israel's kings were quite different of course as they really did have a blessing that was given to them to rule. Other kingdoms, etc. made similar claims but I don't really believe most of them truly had God's favor on their respective bloodlines.