lol. I lived under the same monarch, QEII, who has ruled for seventy years, and I assure you that I and my compatriots feel far, far, freer in practice and guided more well than any single American ought to feel currently.
That apart, the "show" function is important, too. It separates the Head of State and Head of Government functions, which makes it easier to dismiss the Head of Government, without disturbing the stability of the State. The US system fuses the two, which (in my view) is part of what makes the Congress reluctant to follow through on impeachment. (To avoid recent examples: had Bill Clinton been (treated like) a British Prime Minister in 1998 he'd have lost a no-confidence vote among his own party, and Al Gore would have served out his term. I think that would have been a better outcome than what actually happened, and followed.)
Missed that part in your parent reply originally, you're right of course, cheers :)