Changing an icon may cost a bucket, next to nothing, although monarchies are usually very opaque (regardless of what they pretend, we still don't have an account of all European monarchies properties and investments). However, the entire coronation pantomime is bizarre in a country still suffering from the Brexit fiasco.
Everywhere else, I'm sure that may be the case. Except Norway. And Thailand. Maybe Bhutan. etc.
Most countries where a referendum has been held (particularly in the last 100 years) have decided to go with an elected head of stated, not a monarch. (https://en.wikipedia.org/wiki/List_of_monarchy_referendums)
From what I can tell, in Norway it does seem that things have changed only a little bit.
Note I'm not a particular fan of any monarchy, but you did state that "I think the vast majority of people in any country would prefer a Head of State that is chosen by popular vote rather than by Royal Sperm." That's an easily falsifiable statement.
Australia barely accepted keeping the former Queen as head of state the last time this went to a referendum, and that was a member of the Royal family who served in WWII driving ambulances in London and stayed a steady constant since.
It's questionable whether there'd be any love for King (formerly known as Prince) Big Ears should it come to a vote again.
The part that is strongly liked is the notion of something constant and greater than day to day government, a sense of continuity of country.
Day to day meddling in details is strictly for the people we elect and hire to debate policy and pass that on to others that enact it.
So we still have a “constitution” that’s simply a law passed in Westminster.
I despair that the country will become a republic in my lifetime.
but I don't "despair" .. the last time UK lackeys meddled in AU affairs ( The Dismissal : https://www.thedismissal.com/ ( bless the little bleeder ) ) goes back a ways and didn't go down well.
Should that kind of stunt get pulled again it'd likely trigger a Republic.
Technically you don't even need to be on the ballot in some countries to become the prime minister. Granted it would be weird, but ministers are appointed by the party or parties in power in parliament, so there's no rules that says that any minister needs to be elected or even on the ballots. It happens not to infrequently in parliamentary countries that ministers are pulled in from outside. Normally they'll the run in the next election, otherwise they'll have no voting power in the parliament.
I can see the argument that it's not democratic, but it avoids the issues of a presidential election as we seen in the US. In the end, the prime minister is "elected" by the candidates the people voted into the parliament, it works best if you have more than the four parties you seen in the UK.
For those that will vote for him in the upcoming election (directly or indirectly), they still do not want him to be president either.
The monarchy is an silly institution, but there’s something nice about having a (mostly) apolitical symbolic head of state. People look at it and think instead of Liz and Charles, we could have had someone like Trump, Macron, or President Boris, and that scares them off republicanism.
Germany is perhaps a bit better with their elected but mostly irrelevant president, but nobody even knows who they are, and in the end the prime minister acts like the head of state.
The most absurd thing about the monarchy is that it does its job quite well.
I think you are very wrong about that. The Danish royal family remains the head of state because no one want any of our politicians to become president. If feels like a nice safety net to have the king above the prime minister, not an effective one, but still. Politicians are short term, think short term and fickle minded, at least the royal family thinks generations ahead.
As long as the royal family is well behaved, polite and appear interested in the people, I think most countries will prefer to keep them as the official, but neutered figure heads.