The battle of Thermopylae is arguably one of the most well-known ancient battles. People keep saying how Leonidas and his 300 heroic Spartans (there more troops from other Greek city-states at Thermopylae, but they are often overlooked) saved democracy and in effect the entire Western civilization in that battle. The more I read about Thermopylae, however, the more I doubt its significance in term of the second Persian invasion.
No matter how you look at it, Thermopylae was a heavy defeat for the Greek. They intended to hold the pass, not merely delay the Persians. In the end, the Persians were delayed for a few days and the warrior king of one of the strongest Greek city-state, who was also the leader of the combined army, was slaughtered along with his elite troops. Given the Persians' superiority in number, I doubt their loss at Thermopylae bothered them much. With Thermopylae lost, the Greek had to withdraw from Artemisium, because their strategy required them to hold both locations.
In the end, what did the Greek gain from Thermopylae? Did a few days delay matter that much? You can argue that it inspired other Greeks to stand up against the Persians, but is there actual evidence for that? Did the battle really helped boost Greek morale at that time?
Overall, I was led to believe that Thermopylae actually did not contribute much to the eventual Greek victory. In fact, honor and traditions aside, I question Leonidas' decision to keep fighting when the Persians found the secret path. Leonidas did know the situation was hopeless, given that he ordered the bulk of the combined army to retreat. Surely his leadership and his elite troops could have serve the war's cause better in a different situation?
Is there something I missed? Was Thermopylae really that significant in the grand scheme of things?
No, Thermoplyae was absolutely a major strategic victory, and potentially quite significant--just in the reverse order that it is sometimes understood.
That is to say, it was absolutely and undeniably a victory for Xerxes, and a necessary one for his invasion to continue. Remember that Xerxes was a usurper, and one who spent much of his rule putting down endless revolts across his wide, diverse empire. Forcing the pass quickly was of the utmost importance--he could hardly afford to be held up indefinitely by entrenched defenders at an ideal location like Thermopylae.
In other words, it could have been a legendary defeat for Xerxes, but it wasn't. The Greek force was simply too small, though it fought admirably. Even at the time, this was well known. If you read Herodotus, the primary source for most of what we know about the battle, he addresses this directly. Assuming he speaks for the Greek elite's understanding of the conflict, it was indeed a defeat.
But that's the strategic element, and not actually what Herodotus praises about the fight. What made it memorable was the bravery of the Leonidas' rearguard. The tiny army did fight well until defeat was certain, and then Leonidas died in exactly the right way for a warrior-king. That is what was praised, not that the battle itself was a victory. Perhaps modern culture and later writers have somewhat twisted that dynamic, but to the Greeks of the time, that was the moral. To thus answer the question directly, no, the Greeks didn't gain anything from the fight. The delay did not matter, no. It was a strategic defeat, and they knew that. Ultimately, they did take heart from Leonidas' example; he did become quite a heroic figure. But it's impossible to quantify if that really translated to anything concrete in terms of the war itself.
However, it was seen as a victory for the Greek character, which mattered enormously to them. Politics were highly personal in this period; in fact, almost entirely so. Greeks absolutely believed that character was destiny, and seeing a major figure choosing to die fighting was a thing of pride, to be sure. Again, we can't say this 'gave morale' per se; and it certainly didn't reverse the strategic defeat. But they were genuinely proud of Leonidas' actions that day.
If you want to read more about this, I recommend going to the source: Herodotus is the primary source for the Persian wars, and doesn't actually require a huge amount of secondary analysis. I think you'll find that while he exaggerates numbers wildly, like most chroniclers, he shares in your more nuanced appreciation of the battle.