The monarchy might not last much longer if this is indicative of Charles' style.
The idea that the monarch's style, or his impact on public opinion, is of any importance whatsoever to the institutions of government in the places where he is recognised as king, represents a deep and fundamental misunderstanding of what monarchy is, and how it operates.
The tabloid press, who are used to using the blunt tool of public opinion to force changes that they like (and to block those that they don't), are trying to give the impression that public opinion matters here. They are mistaken.
The people, lulled into a false sense of egalitarianism by seventy years in which changes in government have been largely subject to democratic process, and always beholden to at least some degree to public opinion, are apparently expecting that their opinions will, at least to some tiny degree, be taken into account. They, likewise, are mistaken.
It's a monarchy. The king does as he pleases, and everyone else says "How high, your majesty?"
I suspect that the reason that republicanism is not more popular, is that most crown subjects genuinely don't grasp that they aren't living under it already.
The king will not be overthrown because he's rude to his staff. That's not how it works. It's not even close to being how it works.
All that talk of "Charles isn't as popular as his mother, so he may never be king", and "When the queen dies, maybe we will become a republic", was only ever hot air;
Public opinion isn't relevant to monarchies.
And King Charles III is the best of the Kings Charles* so far, by a long chalk. Though that's a very low bar.
*I think the plural of "King Charles" ought to be "Kings Charles"; cf. "Governors General". No appeal to facts on this matter will be considered.