And what self-respecting American would fucking do so? Personally, I cannot imagine being in a relationship in which I would have to literally bend the knee to a foreign head of state in order to stay together. I guess a lot of women are really attached to the whole "princess" fantasy though. "Our whole national identity of not bowing to a ruler? Fuck it, I want that tiara. Small price to pay."
Meghan didn't really believe she'd have to do it long term - her ambition was to "modernise" the Royal family. She ignored loads of royal protocol like skirt length, hats, not wearing all black when you're not in mourning, makeup (etc). By choosing to aggressively pursue Prince Harry, she was opting into being a part of the family of the foreign head of state and potentially the mother to future heads of a foreign state.
You actually don't have to curtsey/bow when meeting royals in general - I've encountered working royals before and you can just be normal at them. Foreigners generally won't be expected to either, the Queen told Michelle Obama that the whole thing was rubbish; the firm expectation basically died off in the Edwardian era. Royal protocol only really engages when you're at a specific royal event like snagging a ticket to the Palace garden party or getting an OBE. The ones who take that seriously are generally of the rah class, who studiously study Debretts and paint their nails "ballet slipper" pink. If you yourself are part of the aristocracy (and so have a vested interested in upholding those traditions) or you want to hobnob the royals, you make a show of following traditions. Kinda like American military people saluting their superiors and referring to them as sir or ma'am, but a random member of the public isn't required to go around saluting soldiers or saying "thank you for your service". Likewise I don't think you actually have to call the President "Mr. President, sir" and could just call them by their first name, although you probably shouldn't if you're trying to curry favour. Or if you work in retail, calling customers sir or ma'am, which comes across as very old fashioned here.
Every culture will have traditions that seem weird to some outsiders, whether that's bowing and curtseying, never passing food between chopsticks, only complimenting babies when you are holding them (or otherwise calling them ugly), never pointing with your fingers, never leaving the office before your boss, kissing on the cheeks, using a different form of "you" to avoid overfamiliarity, avoiding honorifics to avoid overformality, engaging in small talk with strangers, avoiding small talk with strangers, not using your left hand, not touching people of the opposite sex, not tipping because it's an insult, tipping because it's an insult not to, burning worn out flags, making your children pledge allegiance to the country every morning, not wearing white after Labor Day (etc etc). That's the thing about other countries, they have their own culture that is different from yours.