Maybe the USA should get a king?
The USA threw off its king almost 250 years ago, and hereditary monarchy is of course a silly idea, and the remnant of an evil idea.
Yet the King that the UK, Canada and various other countries have is not without value. But there can be much more value that what you may be thinking of, namely saving the actual head of government from ceremonial work.
In particular, a non-political person or body with limited but important power can provide one of the "checks and balances" necessary in a working democracy. Though it's not easy to find the best structure from which to do this.
In the USA, the Supreme Court is supposed to fill this role. They are appointed for life and given a lot of power. Over time, they have grown that power beyond its original scope, and due to a clever 40-year manipulation of the abortion issue, they have also now become far more political than intended. The best path to fixing that may be to explore ways to prevent that political shift, but it's worth looking at how things go in other countries.
King Charles III holds a surprising amount of power, in theory -- a remnant of the way it worked for his ancestors. Laws are not in effect until he signs them. He can dissolve parliament and call an election. He decides who will form the government after the election. Except he doesn't do any of this. He just swore an oath to parliament to obey it. If he were to break it, and actually use these powers, it would become very likely that the monarchy itself would end, and there would be a republic.
Or it may not. In the 1975 Australian Constitutional Crisis, the Governor General, who sat in proxy for the Queen (but is appointed by parliament) used that power and fired the Prime Minister who had appointed him, handing the title to the leader of the opposition, who then resolved a logjam in his favour, and called an election of both houses which he won, handily. There was a great deal of controversy, much protest against the Governor General and some minor constitutional amendments, but in the end the result was accepted, and Australia kept the monarchy. Many felt the unprecedented use of royal power had worked.
Christian X of Denmark also dismissed the government in 1920, in this case over a somewhat political issue. He tried to appoint his own PM but was pressured to just call a new election. This has never been repeated.
The extreme theoretical power of the crown is too much, but today we might imagine there could be value if there were a non-political person or body who had the power to fire the President or call elections if he or she took extreme extra-constitutional actions, including being involved in an insurrection or attempts to corrupt the elections themselves. Impeachment was intended as the means to fire the President for such, and Trump was impeached for such actions, but again, it has become highly political and the process went nowhere.
One of the great challenges is assuring the Crown (not necessarily a man, of course) is not political. The UK has managed to pull this off, and Elizabeth maintained a tradition of staying entirely out of that fray. It's not clear how to duplicate that, since of course people have opinions, and will find ways to justify to themselves and others than their actions, while driven by those opinions, are not political. For Elizabeth, the fear of being the one monarch to destroy the monarchy was enough, combined with her detailed upbringing and devotion to history and duty.
Other systems of rule
Historically, monarchy has been hereditary, though the governor general system in Australia, Canada and other countries is not. It's worth considering some other historical systems which, while flawed, have some advantages not found in European "eldest son" monarchy.
In the Arab world, for example, there is a royal family. The Emir will have many sons (he gets 4 wives) and it is not automatically his eldest son who rules. (The daughters are sadly ignored.) The family has a council which decides which brother will do the job. Not only do they appoint the Emir, they can also remove him and appoint another if he goes nuts. This is still absolute monarchy, which is an abomination, but it's better than the European one.
All the sons are trained from birth to take roles in government. The ones who don't become Emir will take other senior roles, though some may drop out or fail to live up to their training.
A similar pattern is seen among mega-wealthy families who control the modest number of long-lived private companies. These companies were founded (sometimes over a century ago) by an entrepreneur, and some are on a 3rd or later generation of family control. All the children in the family are given a superb private education, with the best quality tutors. The family office works to bring up and train each new generation in this way. All the heirs have stock and they vote on which of them will be CEO. (They can also vote to bring in outsider CEOs, which does happen, and they certainly bring in many outsiders to take other management roles.) Those with talent, combined with their special training, usually have roles waiting for them in the company. Sometimes they get these roles without the merit, of course. And others reject the family company and go out in the world.
Some of these companies fail, but there are lessons from the ones that have persisted, particularly when it's clear it wasn't luck.
China has built the world's largest economic boom using its meritocratic approaches. Even when they had an emperor, doing well in exams was the way to get ahead -- so much so that government work stole many of the best people from the private sector. While China is authoritarian, and ruled by the party which is ruled by its central committee and the leader, to get into the Party you must excel in school. Everybody knows from a young age that to get ahead in that society you want to join the Party, and the Party gets to pick from the best and brightest. Only in the last few decades has business success become the most attractive path, though we've seen that even Jack Ma could not drift too far from what the Party wanted. In spite of the repression within the Party, and the inability to differ too much from the party line, China has recently had incredible success.
It is in fact sad to see how in the last few decades, with two countries of a billion people, the autocratic one has far outperformed the democracy. This does not mean autocracy is better (nor that India has been a well functioning democracy) but it does mean there are lessons to be learned
How to use these ideas?
No major country is likely to revise its constitution around these ideas. It's possible new countries might explore different ways to create checks and balances. Democracy itself is at a crossroads, with its traditional forms under threat from populism magnified by social media and weaponized AI-driven propaganda. It remains "the worst system except for all the other systems" yet may need adjustments and tweaks and reexamination of how it works and its core values. The core value -- that the government is fully accountable to the people -- gets interpreted in many ways around the world, including what other things the government is accountable to (laws, courts, even kings) and how it's chosen. It's worthy of intellectual exploration. Which is all this is at present.
One could imagine a nation where there is an official educational program to work at high levels of government, including being an elected official. This might involve a course which a person must take, as a "minor" in college, in order to qualify for office. (You prefer it to be a minor as we still want people with education in other fields in these roles.) Today, it's rare to get such roles without a university education, most often in the law, though it's not universal. In many cases parents would decide they want their kids to be able to join this "ruling class" -- where "class" now refers to an actual classroom rather than inherited title. People might start young, as the monarchs above did, learning about constitutional values and training in impartiality and critical thinking.
The parties would try to corrupt it, to make sure that lots of their own children enrolled in these classes, and that will have some effectiveness, but children also show good immunity to obviously induced ideology from their parents.
As a side benefit, this might reduce the number of celebrity politicians. Often somebody rises to political prominence because they were famous for entirely different reasons, and name recognition is a strong advantage in a political fight. This is not just Donald Trump, but people like George W. Bush, Jeb Bush and Hilary Clinton who clearly got to their positions only because they had a famous relative. Some of these sort do OK, but the overall track record is poor.
It is likely that such a system could swing officials to be from wealthier families and dominant ethnic groups. It might be necessary to make specific effort to balance against that. But if the body of people who get this degree is large, it will be possible to find many members of each group within it.
The Crown might be a person, or it might be a council, and it might require a supermajority or even unanimity to exercise its power to replace the government. It would not be allowed to do it again for a while, so that it can't choose the government, only remove it. Ideally there need to be consequences for misuse of the power -- checks and balances on the Crown, but necessarily limited to assure its independence. It remains a great challenge to figure out a way to stop the parties from trying to make it another partisan role as was done to SCOTUS.
It should be noted that the Australian governor general was appointed by the PM that he unseated, so even the most partisan of appointment methods still did the job.
Having no election schedule has interesting merits. In the UK, Canada and other UK-derived systems, the government decides when to call an election, with a maximum interval of 5 years. Of course, this is often used for partisan benefit -- call an election when the polls are in your favour -- but the electorate has been known to respond cynically to an election called for just that reason, and vote out such governments. In the USA, which has regularly scheduled elections, the election cycle becomes almost eternal, with at least 2 years of electioneering for Presidents, and no time away from it for house members. This makes elections vastly more expensive, time consuming, and distracting. Canadian elections involve an intense 60 days, and while there is some electioneering outside that, it's vastly less.
This suggests a Crown whose only power is to call elections (independent of the desires of the government) could be interesting. There would be minimum and maximum intervals, but with new technologies to make elections cheaper and easier to run, it could be tolerable to have them a bit more often.
Many other approaches are also worth exploring.
It should be noted that limited Crowns of this sort can also backfire greatly. Hitler didn't win a majority, but Hindenberg, acting in a manner similar to a Crown, appointed him Chancellor. Mussolini came to power by convincing Italy's King and nobility that he could protect them from the sort of communist revolution that had executed their Russian counterparts. Care must be taken to block these outcomes to be sure.