And they did fail at that, leaving Greek cities to be taken by the Persian army, including Athens

Sparta didn't fall though. And anyway, surely they did the best such a small force could do, since they all perished. That's kinda the opposite of failure, no?

The UK didn't fall after losing a good chunk of their army and almost all of their heavy equipment in the Battle of France. It was still a defeat.

In the Thermopylae case, they failed at their intended objective (defeat the Persian army on land) and wasted a good-sized force. The fact that the defeat was set in stone the moment they chose the place and time of the engagement doesn't make it any less of a defeat.

(Note that "the 300" is only counting the Spartiates i.e. citizens; there were a total of 1-2K Spartan soldiers, and many more forces from other Greek poleis.)

A 100% casualty rate is by any definition a 100% failure.

It can be a success if you fulfill your strategic objectives (see e.g. the American units that were nearly completely destroyed in the early weeks of the Korean War buying time for troops to arrive at the Pusan Perimeter). In this case, they unambiguously failed at those strategic objectives.

It’s very clear in Herodotus that a delay wasn’t actually the mission, they intended to win. It just wasn’t a good plan.

Anyway, even if you grant that they were just trying to buy time (they weren’t), dying to the man in exchange for an extra day or so accomplished nothing militarily.

That was not their mission. No contemporary source says so, not even the pro-Spartan ones.