The issue with the opinion polling plot device isn't that public opinion can be manipulated, it's the idea that public opinion carries enough weight to force a change in the head of state in what is essentially a feudal monarchy.
Do you know what a Noble or Monarch that nobody obeys is?
a noisy beggar. That's what they are. It's the followers that make dictatorships work, if nobody obeys your orders it doesn't matter who you're related to, where you were born, or what the law says.
Only someone with followers, can be a ruler.
Public obedience, therefore, matters, they don't have to like you, but you're not in charge if they won't do what you tell them to. If enough of the population tells you 'no', then you'll find your soldiers will also tell you 'no'-and they might kill ya in favor of someone who can tell them to do things, and they actually
do them.
ask the people of the Jacobite period. there was a King nobody obeyed, and a different king, invited to be the king, (Duke of Orange, from the continental europe) whom enough people obeyed to take the throne and there's some doubt as to how legitimate in LEGAL terms that was-but not in the terms of the most basic thing: People did what the Duke of Orange told them to do, and the stuart monarch couldnt' get enough people to do what HE told them to do.
Thus, why it's not "House of Stuart" on the English Throne today.
Obedience is the mark of legitimacy when you strip off all the cosmetic feel-good-about-yourself rhetoric. If the people won't obey your monarch, supreme leader, emperor, dictator, generalissimo, whatever-they're not going to truly be legitimate.