Of course they do. After all they are indoctrinated in its favor from an early age.
I don't see what rational people, starting from first principles, would agree to a bunch of do-nothings sitting in a palace, aristocrats owning huge plots of land that just got handed down to them, that aristocratic class having huge say in politics, networking their kids everywhere, etc.
(As for the "it's good for tourism" argument, you can have your palaces and castles made into tourist places even after you have thrown out your royals. The Palace of Versailles makes millions in tourist fees every year for example.)
You could argue that for almost anything that we have in society (Religion and the State being the two most obvious ones). And yet, it intellectually doesn't phase people the slightest. "So what if we're taught that government is good and benevolent since a tender age". This is precisely because most of society does not function using first-principles. If they did, we'd be in a completely different world right now.
Is that supposed to be an argument in favor of royalty? At best, it's about how other things people think OK and are used to from a young age are also messed up. One could use that line of logic to excuse all kind of societal issues at their time -- segregation, racism and sexism for example.
And yet, society evolved beyond those, from self principles like equality.
So? The US also wanted to do the same (build a society based on "certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness"). The French too during their revolution. As did countless other revolutions besides the "commies".
Didn't end that bad for most of them, in fact that's how the modern world was build.
Would you rather we still had Kings and Pharaohs lest "some people turn to be more equal than others"?
Why is your elected president hijacking the telephone network worse than your unelected queen hijacking your tv network?
Also, if you switch over to Channel 4, you can watch an alternative speech. A few years ago, they had the President of Iran on simultaneously.
Currently, a little bit more of the half of the population supports Erdogan and the other half hates him. Though admittedly the anti-Erdigan half did not support the coup as it is believed to be orchestrated by another islamist, but still...
So, just like most politicians?
My point was that the British public would not be happy if the Queen used her time on TV to push the agenda of the half of the nation - like openly supporting or opposing Brexit using channels that are not available to the opposition.