[Warning: None of this is remotely related to any area of expertise that I actually possess.]
So I've long been annoyed by monarchies — including the most milquetoast Western European constitutional monarchies — just on general small-r republican principles. My objection, when it comes to the liberties and welfare of the people, is almost entirely symbolic and aesthetic, given that the monarchs don't really -arch any more.
But I'm also troubled by how monarchy distorts the lives of the Royal Family's children. It gives them unfair advantages, of the sort that can be harmful as well as helpful. But it also imposes on them unfair constraints. They may well be limited (perhaps legally and perhaps by family honor) in whom they can marry. They are limited by family obligation in what they can say, what causes they can champion, what jobs they can go into, and so on. It's very hard for them to lead any sort of private life in which they are judged by the normal standards applicable to ordinary people.
And this isn't something they voluntarily chose; it's thrust upon them, for their whole lives. This is in some measure true for younger children of democratically elected politicians as well, but my sense is that it's much more true at least for the British royals and, I'd guess, for many others as well. Hereditary privileges, obligations, rights, and constraints are, I think, unfair and potentially destructive to their holders.
Yet I have to recognize that it's hard to dislodge centuries-old traditions, such as the monarchy, that are emotionally important to citizens. The traditions can change, even quite substantially, as they have with regard to royal power and royal constraint. But simply shifting from a monarchy to a republic, with no reason other than aesthetics or a worry about the welfare of the royal children, is probably too much.
Hence, my humble proposal: Why not retain the monarchy, but (1) stop its being hereditary, and (2) institute a practice through which the figurehead monarch is chosen by Parliament based on his or her great accomplishments during his or her long life (preferably at least 60 years or so)? The offer to Albert Einstein of the Presidency of Israel (an offer that he of course declined) might be something of a model, though you'd expect Presidents to wear socks. The advantages:
The titles, trappings, and most other incidents of monarchy will be preserved, but the symbolism of royalty will become recognition based on great merit, rather than hereditary right.
I expect the affection of the people for the office will also be preserved, since I take it that people who love the Queen love her chiefly because she's the Queen and not because she can trace her lineage to the Electress Sophia of Hanover. (I recognize that some love Queen Elizabeth II is because she's been queen for so long, which my proposal wouldn't provide for; but I take it that many other Kings and Queens reign for less than Queen Elizabeth II has, and that the length of the reign is not the main determiner of the nation's affection for the monarch.)
The tangible and symbolic benefits will in fact be given to those who merit them, and not based on accident of birth.
The King or Queen will be someone schoolchildren will have good reason to admire, and that foreign dignitaries and others may actually be independently pleased to meet.
The selected person will be a known quantity, so people with bad character — or even personal habits that might be perfectly fine in a private citizen but might not be optimal in a head of state — can be screened out up front..
The selected person's family will still be delighted, but in most situations the children won't even have to grow up in the shadow of the parent's office (since the children will probably already be adult by then), much less feel substantial constraint on their lives from the parent's position.
Because the person will be near the end of their career, the potentially time-consuming ceremonial duties of royalty will probably not take that much away from the value that the monarch could contribute in the science or art that brought him to the throne. (And wouldn't it be good for the worry to be that having your King be King will make it harder for him to make still more great discoveries or creations?) But the monarch's elevation might make the monarch an effective spokesman for more private contributions to that science or art.
Now that the monarchy is largely powerless, the historical objection to elective monarchy and in favor of hereditary monarchy — that in each election so much will be at stake that the nation will come close to civil war — will obviously not be available. While there might be some behind-the-scenes dirty politics in such elections, as in any political endeavor, the practical peril posed by such politics would be minimal.
So, our British, Dutch, Spanish, Swedish, etc. readers — are you with me?