It shows. The Republic is not a political treatise. Everything in it about city-states, philosopher kings, etc., is an overarching metaphor. He states multiple times throughout that it's not a realistic political system, would be impossible to implement, and wouldn't even be desirable. From the outset it's framed as a device to illustrate a point about how individuals should compose themselves in relation to their three-part natures: rational, spirited, and appetitive.

In modern terminology we could call it the intellect, the emotions, and the body/instincts. The whole spiel about the philosopher kings is merely to say that a person's rational nature should be in charge of their emotions and physical drives. This isn't interpretation, he literally states it plain as day multiple times.

Although he does make a point that someone who's thoroughly educated and has a strong reasoning capacity would make a better (more just) ruler than someone driven by hunger/greed/lust/whim. But that's more of an aside. And he's not exactly wrong, either...

Anyway, the context surrounding the Cave analogy was basically him making the point that everything we experience is more or less a hallucination represented by the shadows on the wall. It's a surprisingly accurate intuition, given what we know now about psychology and perception.

Coupled with the Cave analogy is another analogy where he describes a spectrum between knowledge and belief. For Plato, "knowledge" was a very precise term, and much of what we think we know is merely belief. Anyway, the Cave is basically about how we all tend to live in belief, mistaking it for knowledge, in the same way the cave dwellers mistake the shadows for objects.

To put it another way, much of what we view as objective reality is actually highly subjective. Even the most concrete, systematized models of understanding reality, at their core, are ultimately founded upon basic assumptions.

He also talks about people who have left the Cave (ignorance) and seen the world above in broad daylight (knowledge). When they return to the Cave, everyone thinks they're crazy and they get shunned because nobody wants to hear their crackbrained ideas about "sunlight" and "trees," so they become a pariah.

And if you've ever tried to explain climate change to a denier, then you likely have some experience of what it's like to engage with the ignorant from a standpoint of knowledge.