I've always said I wouldn't bow for anyone either, that's just my opinion. I know of abuses in a country called Canada in which certain agencies and their agents believe they are God and it has ruined us immensely.
Look, we are all from the same Creator. I don't treat the janitor or fast food worker any differently than I treat the CEO of a corporation. Yes, I'm respectful, I call police sir if pulled over, I give people respect where warranted, but to me, personally, bowing is not in my DNA (well technically it is, but not me to follow the Old Country).
I wouldn't have faulted Trump for bowing if he decided to, but the fact that he didn't says "I'm an American now". Americans believe in equality of men and women and this probably plays well with Americans quite frankly.
Again, I don't have anything against the Monarchy per se, it's a personal view.