The rationale goes like this: If you don't like the monarch, who has this job purely as an accident of birth, well, too bad, they're a person, that's all you were guaranteed. They might have a grating voice, or love hot pink way too much, or insist on calling people "Mate" for no god damn reason, but nobody picked them.

And if you do like them, the same, maybe they've got a brilliant smile, they like the long floral outfits you prefer, they think kittens are better than puppies like you do, but nobody picked them.

They don't have any democratic legitimacy, which is crucial because this means it's clear that democratically the elected legislature is in charge. The monarch can't be making rules, they're not a government they're just a living symbol.

Whereas if you vote for somebody to be President or whatever, that person now has democratic legitimacy. Their preference for kittens is granted the legitimacy of the masses, even though probably most people didn't vote for them because they believe kittens are better, too bad, that's all endorsed. If the Elected President disagrees with the Elected Legislature, somehow their meddling is legitimate, even they can argue, rightfully endorsed by The People.

Unless you're confident that everything you believe can be summarised as a single living individual, and that the majority of other people agree with you, an Elected President is clearly worse than a Monarch.

Well, in the US, Supreme Court appointments are for life. That's for a reason - it's supposed to be isolated from politics so as to judge justly. It's not, but it would be even more subject to politics if judges were appointed to four-year terms (or worse, elected).