I live in a constitutional monarchy, where all the functions of government and state administration are controlled by democratically-elected (or indirectly appointed) officials, and also we have a monarch off to the side in a ceremonial role. I would agree with the above interpretation.
As I mentioned in my recent thread about the coronation of Luxembourg’s new Grand Duke, there was a minor crisis a few years ago where the monarch refused to put his signature on a law to which he had religious objections. Parliament swiftly assembled and made the necessary Constitutional adjustment to clarify that the Grand Duke’s signature was simply a rubber stamp and he had no authority to block legislation. The crisis was meaningful, but it was also extremely brief. There was never really any question about whether the Grand Duke might retain this power.
It’s a little bit baroque, but it seems to me that the People conclusively demonstrating that they have a leash on their monarch, and not the other way around, is a pretty decisive affirmation of democratic rule.
Of course, you can then argue that the monarch is extraneous, and not “part of” the democratic system. Luxembourg’s royalty could evaporate tomorrow, and it would make no difference to how the country is governed (minus one ceremonial flourish). The rebuttal to this is that the existence of the powerless monarch is itself a concrete demonstration of where the authority actually lies. I don’t know that I subscribe fully to it, but it’s an argument that can be made.
And, yes, I’m also sympathetic to the comment upthread that the American drift toward “a king in all but name” suggests that there might in fact be some genuine symbolic utility in having a toothless thronewarmer on permanent display.