Just read the Republic and see for yourself. Kallipolis is, first and foremost, simply meant to be a metaphor for the organization of an individual's soul. Second, the suggestion that the city should be led by a caste of authoritarian Philosopher Kings is given inside of a conditional---the condition being: those leaders must be True Philosophers; where True Philosophers know the Good and thus know (in a nearly omniscient-like way) what is best for everyone (and act accordingly). It is left wide open whether such True Philosophers even exist, thus it is left wide open whether such a social organization would ever work in reality.

Plato's other political dialogues, like Statesman and The Laws, are much much less utopian and "authoritarian" and deal with very practical political issues. It would be weird to be an authoritarian utopist and then go and write dialogues like those.

I think Popper would object to the phrase "truths are only true if they are verified". We don't knowingly verify truths. The things we think are truths aren't even true, they're just not false (yet).

By the way David Deutsch could be reasonably said to be Popper's biggest fan. If you're interested in Deutsch and Popper you could do worse than picking up "The Beginning of Infinity" to get acquainted with both, it's a great book.

The optimistic read of that problem is that the devolution is itself the corrective. Athens outpaces Sparta through openness, devolves, falls. Rome's republic succeeds it, devolves, falls. Promoting growth is then: promote openness as a principle and accept that when a society devolves, its failure clears the way for selection for increased openness.

Otherwise, it’s important to remember Plato sock-puppets Socrates, who had no truck with the newfangled and subversive invention of writing and thus could not correct the record. What is clear is that Plato, a disgruntled aristocrat himself, exiled for being part of the quisling faction, was a proto-fascist far beyond the wildest dreams of a Stalin or Hitler. But philosophy teachers like the conceit of a philosopher-king and that’s why he hasn’t been consigned to the trash heap of history where he belongs.