I feel like there are several things to be addressed in your question.

First of all- slavery in the ancient world is not exactly the best way to judge whether or not a civilization should be considered developed as for most of the ancients it was universally accepted. Even Solon's reforms in Athens that allowed for the "shaking off" of financial burdens and freeing of the formerly middle-class was not a movement away from slavery, but more so an endorsement of the independent farmer's financial viability. Though slavery was certainly bemoaned on an individual level, the concept of owning human beings was not viewed as an invalid social construct until long after Greece, Persia and the other eastern empires you referenced were long things of the past.

Though perhaps not "uniquely" civil or high minded, the Greek achievement should not be minimized. A few examples of things that set Greece apart and account for its continued eminence:

Homeric Epic- The composition of the Iliad and the Odyssey through oral tradition in the archaic age, as explained by the Perry-Lord hypothesis essentially says that these epics cannot be owed to a single author, but to a group of bards, working within a pre-literate tradition over decades. Though "illiteracy" is often seen as a mark of underdevelopment, the Greeks developed a huge amount of rich culture and national identity before inheriting the their alphabet from their Phoenician trading partners. The gravity of Homeric Epic cannot simply be explained by its effect on later western literature, though. The narrative complexity, literary cross-referencing, and perhaps most importantly, cementing of key canonical lore can compete with anything produced in the eventually literate world of the next few centuries. In regards to your question about influence- the epics are an excellent example to look at for that as well. During the early colonization movements, the oral traditions cemented in the epics served to code for societal expectations. Through showing causal relationships between appropriate and non-appropriate behavior and divine response, the epics that traveled to distant colonies preserved the citizenry's "Greekness" through these representations of Greek culture. As explained in M.I. Finley's book, The Age of Odysseus, Homeric epic did not so much explain the way a distant Greece worked at the time of their emergence, but rather preserved the complex social constructs of the Archaic age.

Politics- This is a good place to examine what you mean by "a new age of freedom" and the issues arising from the depictions of Greece in 300. First of all, it's worth noting that Ancient Greece was not a single civilization but rather a group City-States that shared a common language and religion. Many ancient authors, Herodotus being the best example, address a quintessential "Greekness" in their work, but politically, they were independent. Athens operated as a democracy, with tiered levels of representation and political activity. Sparta was essentially a military oligarchy benefiting a small minority. However- 300's references to liberty are misleading if we do not try remove our own connotations from that word. Whereas liberty and freedom to Westerners today are concerned with individual liberty, Ancient Greece would have been far more likely to view them as terms applied to the citizenry as a whole. So- when Darius and later Xerxes threaten greek "liberty," it isn't so much their threat to each individual's ability to self-govern, but rather the the preeminence of the Polis (city-state) as the unit of political organization and that community's ability to select how it self should be governed. A critical piece of Herodotus to explain this is Persian constitutional debate. As Darius and his co-conspirators debate how best to govern the Persian empire once they have taken control, we see a beautiful display of the importance Greeks put on a people's decision for their style of government. (That needs a bit of unpacking.) Herodotus is here showing a foreign people's developed rhetoric, oratory, and political self-awareness that there is little corroborating evidence for in Persian records during this time. He is assuming that the choice of a political system for another people is as methodical and self-defining as it would have been in the Greek world. So- the reputation Greece receives in terms of political development is not only dependent on their novel political systems (democracy and the like) but also in the importance of political identity. We see the Greeks as inherently political people because it was such a critical piece of how they chose to view themselves. This is much better explained by Charles Freeman in his book The Greek Achievement. He essentially answers a lot of the questions you raised about just what made Greece a beacon of intellectualism in the ancient world. Conceptions of liberty in Greece, and more particularly anywhere but Sparta, were also dependent on the existence of a middle-class. Whereas much of the world operated through principalities and rigid social immobility, the independent Greek farmer was both a political participant and military defender of his Polis. The famous greek hoplite (again, not discussing Spartan hoplites) is a phenomenon impossible without a middle-class. Champion warfare (that depicted in epic) of people fighting one on one, was largely a thing of the past by the 500's. With each free citizen providing their own armor and banding together to defend their communal land, the importance of the demos or citizenry again becomes significance. It was the duty of the individual to protect both what was his and what was his community's. We can contrast this with the army that Xerxes raised in the mid-fifth century composed almost entirely of mercenaries and subjected people. It is understandable that the reputation of Greek's defending liberty has become so canonical as the average soldier in Greece was fighting for himself and community while the average invading Mede was fighting for the expansionist doctrines of a king.

Power- You raised an incredibly interesting point when you said that Greek was a "small backwater" kind of place. Militarily, the Spartan infantry and the Athenian navy were the prides of the Greek world before the rise of the Macedonians years later. You were also correct in your observation that the eastern empires were larger in their military scale and also in their geographical holdings, but you're mistaken that this necessarily equates with Greece being somehow underdeveloped. The Greek achievement before the age of Philip II and Alexander was absolutely not one of military expansion. Though the victor of both Persian invasions, a desire to expand as a unified Greece seems to play very little in the minds of their leaders. The Greek military was powerful, and a huge amount of popular focus has been placed on that in modern years (likely because it sells a lot of graphic novels), but the importance of Greece's military conflicts with the east rest on a more abstracted idea. Greek excpetionalism, and more importantly Athenian exceptionalism, refers to the feelings of invincibility, inherent superiority, and cultural dominance acquired through the beating back Persian forces. The tiny states of Greece, ununified, had defeated the largest army that had ever been raised. This alone is impressive, but it also created a "golden age" for Hellenic civilization. Playwrites such as Euripides and Sophocles, philosophers such as Socrates, ethnographers, rhetoricians, mathematicians, all made huge cultural and scientific progress due to feelings of Greece's importance. Essentially- the Persian invasions of the fifth century endorsed Greek progress even beyond what it had been beforehand. Richard Billows addresses this point far better than I can in his book Marathon. The thesis is essentially that Greece benefited greatly from their wars with Persia because it ramped up cultural output. That particular book focuses on Athens in particular as they were the victors of the Battle of Marathon, but most of his writings are pretty solid reads.

Conclusion- Greece had a lot of things that justify its place in history. Misrepresentations in media aside, the political and cultural developments of Greece rivaled the far larger eastern empires and was even able to defeat them, despite economic and population disadvantages.