Plato's Republic is an enduring cannon work that's largely identified with Classical Greek and Western Philosophy (Antiquity).
It introduces several perennial ideas:
I will argue below that The Republic is a Noble Lie justifying The Philosopher King. This follows from the failure to cross the Epistemic Access divide (failure of justifying The Form of the Good) - something demanded by the strictures of the Socratic Method and right alignment of the Tripartite Soul (particularly within a philosopher).
Consequently, The Republic, by its own standards, collapses into Sophistry, mere Appearance of justification over the reality of such. Thus, the Form of the Good concept raised in the book collapses into mere Sophistry and reveals itself as a Noble Lie.
According to the Socratic Method, an argument or dialogue that ends in contradiction is to be discarded as untrue, unworthy of a philosopher, irrational to believe, or false. (Introducing basic notions of Soundness, Validity, and non-contradiction.)
Simultaneously, ideas must have justification (reasons) in order to be substantiated and confirmed in an argument.
Lastly, the Philosopher (King) is to be rightly aligned with and dominated by Reason (the Tripartite Soul).
A variety of criticisms against the primary (metaphysical) thesis of the work (and often considered the most significant and enduring idea) are raised and dismissed within the work purporting to justify the existence of Platonic Universals that derive (in some sense) from Ultimate Form of the Good.
However, as I noted elsewhere, no argument is given to justify the distinction between so-called Abstract and Concrete objects (using a modern gloss on the terms - they are presupposed to justify the argument for the Form of the Good in the first place).
Thus, within the text, there doesn't seem to be much-motivating justification for the concept - once we've applied the Socratic Method to the Form of the Good.
More recent arguments challenge the coherence of the concept as well (Bradley's Regress is perhaps the most famous or well-known).
One cannot argue from a weaker position to a stronger one. E.g. - the following argument structure is invalid:
One cannot generally (here: universally) move from existential quantification to universal quantification.
So, one cannot argue from a critique of human society to general metaphysical truth. Indeed, the Parable of the Cave is used to motivate (by way of analogy) but not definitively argue for (the truth of) the thesis that all things are Platonic Universals derived from Form of the Good.
In fact, The Parable of the Cave is an illusory source of justification, the very kind discouraged as Sophistry elsewhere in the book.
So, of:
3. lacks justification since 4. cannot motivate it and collapses under scrutiny by 6..
As such, neither can 7. demonstrably rule in favor of 2. (I won't go into this much here but it should suffice to say that 3. plays an essential role in formulating the ethical component of 5. and 2.).
Therefore, The Republic collapses into mere Sophistry and reveals itself as an illusory Noble Lie (by its own definitions).
These criticisms align with other thinkers like Nietzsche, Aristotle, Rousseau, and Hayek and their respective criticisms of Plato and Socrates.
The structure of the book lends credibility to the claims above. Consider that Socrates is merely the protagonist of the work - it is not a first-person auto-narrative or primary text. It is a work within a work.
Our understanding of Socrates is derived and the book itself is a veritable cave about Socrates.
Historical context suggests that Socrates existence was partly disputed (at that time and in ours - he may very well have been a fictitious person and such a belief was present at that time), the fact that Plato wrote the work about Socrates perhaps further lends itself to the interpretation above.