In sympathy to Popper's descriptions, putting intellectuals in power (via Plato) is how you get tyranies in the first place, as that's what the 20th century was about. These weren't kings with divinely ordained rights, they were ideologues and demogogues who wrote and used books and manifestos as vehicles for populist movements (ironically, anti-intellectual ones), and then established tyranny that iterated the logic of their ideas. It's not restricted to secular intellectualism either, as theocracies are also run by religious intellectuals instead of secular ones.
There are likely equivalently devastating criticisms of Popper, Marx, and others that are as strong as his of Plato as well, but I think their entire goal was to create an intellectual solvent to dissolve the bonds of reason, thinking, and belief, and not to produce positive models using ideals and principle (like, say, Aristotle). (Article states Popper's intent was to unmoor society from it's great man thinkers) It's just a system for producing neutralizing criticisms, which are sufficient for his purposes, so criticisms of those ideas are why they are met with personal and political attacks today, because the theories aren't designed to withstand anything, but only to dissolve others.
The basic error is that all criticisms based on materialism and power are necessarily predicated on an axiom that the good in the world is both finite and zero-sum. Personally, I reject most marxist and progressive critiques and their descendents because of this basic fallacy. Whereas the only zero-sum thing is stuff you have that they're trying to talk and bully you out of instead of discovering or making their own. The redemption they offer in return is subjugation. It's not a philosophy, it's a fancy bandits hustle.
As I remember just from the few readings I did, Plato's ideas were about the city state, which means their scale is limited and you can leave to find or found another one, and not governing the layered federations of city/state/nations we typically advocate democratic rule for. Scaling a republic out of the city state model may have been a uniquely American innovation, aided by adapting the division of powers I think from the federated model of the Iroqois and other native nations. I'd wonder if Popper had used Plato's city state model as more a straw man and a vehicle for these more generally destabilizing criticisms than as a basis for positive alternative.