Glenda's Legs
Paul Walsh
You don't have to bow to royalty any more than you have to respond to someone asking "How do you do?" or thanking someone for a gift or card.
It's just rude not to.
You're right, no one is going to get beheaded if they don't bow. But it is supposedly 'protocol' and therefore expected. I don't see why it would be rude to choose not to bow. It is possible to greet someone politely and respectfully without needing to bow or curtsey.