The toughest part about being an anti-monarchist is that the monarchy's power is largely symbolic. It's difficult to argue that the institution subordinates the people when there is no visible manifestation of that, save the language of the law. But it's discussions like these that give us our answers.
See, there are rational sounding arguments in favour of the monarchy, but most monarchists don't know them. Most monarchists at best will argue "tradition" or at worst feel
their reasons at you. That is to say, they only support the monarchy for some amorphous emotional reason, which can only be expressed in blabbering on about how wonderful it is. It's a feeling I'm sure you might have felt as a teenager (at least I have, though I'm younger than most here). Teenagers get overly excited about things and exaggerate their importance despite not having the words to express them. Their enthusiasm is expressed in how powerfully they can emphasize the adjective they are using, i.e. how much they feel it at
you. The enthusiasm is often lost in communication.
When I read some of the posts in this thread, I detect attempts to convey this enthusiasm for the monarchy in words, but it's totally lost on me, like it's teenager trying to convey how "epic" something is. Like someone explaining some religious principle. It's all white noise. Meaningless babble. But it does demonstrate how the monarchy subordinates the people. Reading the effusive praise heaped on the monarchy and the royal family one bares witness to that subordination. To watch adults babble on about these celebrities, the absurd things they reduce their intelligence to praise, insults their dignity as human beings.
It would be disturbing enough they were just normal celebrities, but in all this they are expressing joy that these people are all rulers. It's odd when you see this kind of enthusiasm in the United States, but at least the subordination isn't there.