I think Norway is fairly unique in that we actually voted to have a king (12-13th of November 1905) after the end of the union with Sweden. Also our king's power has been reduced to a democratic emergency brake where he can only delay a law being passed until there's been an election. If it is passed again, it becomes law regardless. Formally he's the sovereign though, the one signing all the laws, head of all military branches, the one formally leading the king's council with the prime minister as his first advisor. And his person has a total blanket immunity in Norwegian law, though it was settled that he could be sued in a property dispute.
What I find quite appealing at times is that he's not a politician, not looking for a reelection or to further his own career nor is he trying to represent just the 51% who voted him in. In the US I have the impression that if a Republican is in office all the Democrats hate him and if a Democrat is in office all the Republicans hate him. He represents the nation of Norway and not whatever political party happens to hold the reins at the moment. There are other nations that have a form of ceremonial leader like for example Germany with the Bundespräsident as opposed to the Bundeskanzler, but it's a retirement home for politicians. You have to campaign to win it. It's not forever, so there's self-interest to it.
Our king is pretty relaxed about his right to rule, or rather I feel he thinks it's more of privilege. No blue blood, no divine right to rule and I think he like pretty much all western monarchies knows he sits at the parliament's mercy. Like the US, we do have a constitution and a process to amend it and like I said he couldn't block it. If he was losing the people's support I think he'd resign gracefully though long before it came to that. And apart from at the coronation I don't think I've ever seen him with a crown and all that, it's more a ceremonial rite when you take the job.
You can of course say he's not needed, that the US is a nation independent from the President in office and I suppose that's true, but it's a very abstract and silent existence. For example during WW2 the radio broadcasts from the king in excile in London was gathering the nation. When people use archaic expressions like "for king and country" we're not talking about saving one man's divine ass anymore, but that the king serves the country and we follow him as our leader. It's not a perfect system but honestly speaking I feel it works well. It's good for tourism. Sure they live in a castle with a solid upkeep, but I know we'd keep it for historical reasons anyway. We'd no more tear it down than old churches.