I think the Kaiser Wilhelm/Daily Telegraph incident says otherwise. While obviously the consequences in the modern era wouldn't be as drastic as that having a King put their foot in their mouth is still something that would be politically damaging.
Not even remotely similar. Wilhelm said that in an interview, Charles said his thing (which wasn't even saying Putin = Hitler) in a private discussion at a Holocaust museum.
For the record I don't think Charles should be skipped, but I do think his outspokenness (relative to the Queen) could result in a blunder that stirs up support for an Australian Republic in the future, whereas William seems less likely to blunder.
Every king will be outspoken relative to the Queen. People will have to get used to the idea that monarchs may actually be willing to speak publically.
For the record pt II, on Monarchy itself, while I have an aversion to the 'Divine Right' concept, recently I have been thinking it is probably preferable to a Presidency. For one it seems to me to be be far more incorruptible, there is no real incentive for the Monarch to engage in partisanship as not doing so is far more advantageous to their position than any bribe could be. The situation in Sri Lanka does not seem possible (at least to the same extent) with our current system.
I think this kind of reasoning is why Australia will not become a republic. Having a president doesn't look as attractive today as it did 20 years ago.