I am no admirer of monarchy, the royals, the wealthy, or the elite, but modern Britain has made its system work by having a monarch as a unifying national symbol in a country that has experienced considerable ideological and political turmoil. Having a monarch as head of state lets politicians and the people be their usual, scheming, greedy, and foolish selves with a diminished risk of the country coming apart at the seams.
As for a case against the British monarchy, I think it rests most strongly on three points.
First, the British monarchy props up Britain's class system, which offers an endless source of resentment and distraction and justification for cheating by the populace. Why not go on the welfare dole as a fiddle when the country is governed by the great fiddle of the monarchy and the class system?
Another result of the class system is that Brits tend to have a sense of manners and boundaries that contrast with American directness, the idea that the customer is king, and a get it done ethic. That is one of the reasons why Brits find America so appealing.
Here, the idea killer of "that's just not done" is virtually unheard outside of Ivy League faculty lounges and elite private clubs. A common American response to "that's just not done" is why not? If a good answer is lacking, the idea can move forward for discussion of the merits.
Second, the bad behavior of Britain's royals is a major problem. It is often deeply unseemly and gets them into association with shady characters. Prince Andrew's friendship and frolics with Jeffry Epstein is the most recent example but is hardly an isolated one. Again and again, being a wealthy and titled celebrity has the to be expected corrosive effects on character and conduct.
Third, money-grubbing by the royals can have an effect on government favors and policies. It is no secret that if you wish to get on the royal honors list or get a special immigration permit, the way to do it is to spread money around via charitable donations and favorable business deals for both politicians and royals.
The more troubling aspect is when this kind of pay for play extends to money-making contracts with the government and policies that are questionable in their effects on the public interest. Britain's foolish embrace of mass immigration, especially Muslim immigration, was due in part to covert royal influence after the British monarchy entered into lucrative business arrangements with the Saudis.
Similarly, royal investments can be said to have resulted in the monarchy supporting more favorable policies toward countries in the British Commonwealth. The sentimental claptrap about "the Commonwealth" is in part a disguise for a lot of wheeling and dealing over trade and investment.
So, the Queen is visiting a Commonwealth country? Outside of the ceremonial stuff, she likely has a member of her entourage checking on her investments and sniffing out new deals. And if that goes well, the elected British government will get a talking to about the need to improve relations with that country by giving them the trade deal that has long been under discussion.
The British royalty's investments are mostly undisclosed to the public, but the British government has the benefit of intelligence on them. Supposedly, that is one of the reasons why Thatcher and the Queen has a sometimes rocky relationship. Maggie knew that, behind the scenes, Liz was often taking care of personal money interests when she and her minions urged the government one way or another.
Yet it is not at all clear that Britain would do better as a federal republic like the US. Even now, regional antagonisms tend to undermine the unity of the United Kingdom. A further downsizing of the British monarchy though makes sense. In a way, that is what Queen Elizabeth seems to have had in mind by drawing a distinction between royals with public duties and those who have a title but no public role.
We shall see if King Charles is more or less of a dolt than he has seemed to be, and if horse-faced Queen Camilla can put her seedy sleeping around and multiple prior marriages behind her. Who knows, but the House of Windsor's royal minders and rule book may manage to make them into respectable figures pending the accession of the genuinely popular William as king.
Enoch Powell was a prophet