[eszed]

They don't have power, but they do have influence. Every Prime Minister has a weekly meeting with the... King (still hard to type that, isn't it?), an off-the-record discussion of what they're doing and thinking, with someone who (theoretically, at least) has the long view interests of the nation in mind. Most have said it's immensely valuable.

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.)