While this isn’t really consistent with the myth of Spartan superman, isn’t it more than enough to explain their reputation in the fifth century?

Yes, at least by the end of the fifth century, when these things are securely attested. That's exactly what I said, though. The second post of that three-parter is devoted to showing what actual advantages the Spartans had, especially from the middle of the Classical period, and how it gave them a big edge in pitched battle against the untrained armies of other states. I'm not sure what your question/criticism is on this point?

They must have had something going for them to maintain such a successful interventionist foreign policy, no?

Yes, as I said in the other post, they leveraged their numerical superiority, as one of the Greek states with the largest territory, in order to subject one Peloponnesian state after another. Every victory added more men to the levy of their hegemonic alliance system, allowing their power to snowball. By the second half of the sixth century, when they began to cultivate their reputation as tyrant slayers, they already controlled most of the Peloponnese through a system of unequal alliances we refer to as the Peloponnesian League. At the time of the Ionian Revolt in 499, the Spartans were generally recognised to be the "leaders of the Greeks" - the most powerful Greek state and most prominent representative of their cultural zone.

Due to the bad source situation for the Archaic period, we don't know exactly how Sparta rose to such prominence. Many scholars have traditionally connected it to their later military customs and tactics, but there is no concrete evidence of any of these in the Archaic period and nothing to suggest Spartan military superiority. Indeed, Sparta's rise to regional supremacy was bumpy, with Messenia resisting occupation for perhaps as much as a century, and Tegea inflicting a major defeat on the Spartans at the so-called Battle of the Fetters in the middle of the century. Matthew Trundle (among others) has suggested that they introduced many of their more militarist reforms in response to the latter defeat, which would mean an increasingly "Spartan" Sparta from about 550 onwards.

Do we have examples of other, similarly sized Greek citizen armies moving that quickly?

This is a good observation, but unfortunately we have nothing to compare it with. The fact that the Spartans were in a hurry may be enough to explain the achievement. It would be a bit too convenient to use this as evidence for superior military organisation, since the Spartans were not known to be especially quick march hares in later times.

Why were the Spartans given command at Thermopylae and Plataea? Was it sheer numbers?

I already noted this above. Due to the Spartans' political hegemony, they assumed the right to lead the anti-Persian alliance. This was a matter of ultra-competitive Greek morality, and Herodotos describes how the Spartan claim to supreme command made the Syracusans and the Argives choose not to join the resistance; they disputed the Spartan supremacy and refused to serve under Spartan commanders. Herodotos singles out the Athenians for special praise because they accepted Spartan leadership even on the sea, where it should by rights have been theirs, since Athens provided more ships for the Greek fleet than all the other allies put together.