They exist because we agree they exist.
Yeah, I think that's the crux. And I think you (or Wes, can't remember who said it) summed it up that it's about moral weight. E.g. An action taken with UN approval has moral weight because it indicates a widespread agreement that it's a morally correct thing to do. Or, the corollary, an action taken without UN approval carries the implication that it might be morally shady.

That's a bold claim. Your riding an awfully high horse there.
I'm not sure it is that bold. We don't have fixed terms over here so a PM can face a vote of no confidence and a re-election at any time. They don't even need to face a re-election because our Prime Minister role is really just an honorific that also happens to carry authority. Their own party can kick them out at any time based on their own party rules, - it's not dependent on the laws of the land. Our bar to remove a PM is WAAAAAY lower than yours is to remove a President with a whole impeachment procedure. So he certainly could be removed under our system.

Whether he would is more debatable. Trump's approval rating have tanked, watching some of the Town Halls it's hard to imagine that some of your Senators and Congressmen aren't feeling like their positions are threatened, I kinda get the feeling that some of his cabinet have issues with Musk's antics and the discomfort on Rubio's face in that Zelensky press conference looked pretty stark. It's hard to tell because those people will present a united front, whether they believe it or not, because you're system makes the President massively less assailable than they are. Under ours, where a PM faces more of a threat, his party is more likely to break ranks because, by inference, they are less threatened.

I think YerEverLuvin is probably being a bit hyperbolic citing two weeks but he's definitely not beyond the realms of possibility. Over here you would probably at least be hearing rumblings of dicontent right now.

All that said, our PM isn't really the equivalent of your President because our PM isn't the head of state. Out equivalent would be somewhere between the PM and the King. We have removed Monarchs in the past (Charles I is the obvious one but there's also been the Glorious Revolution and various historical usurpations) but our bar to do that is way higher than your impeachment.