One of the sides looks like it's been surprised and is rushing out to battle piecemeal. The actual combat, however, looks pretty clearly to be between two ordered lines of heavy infantry, not two disorderly mobs with heterogeneous equipment.
And I don't think it's accurate to describe the soldiers depicted as "missile troops", either, since they're clearly equipped for close combat with heavy shields, body armour, and enclosing helmets. They seem to have javelins, but then so did the Roman legionaries; carrying javelins into battle isn't incompatible with using close-order, close-quarters combat as your main tactic.
Herodotus says that Cyaxares, the kind of Media, "first ordered the men of Asia into companies and first arrayed the spearmen and archers and cavalrymen apart from one another, whereas before this they had all fought mingled together without distinction" (1.103.1). This doesn't at all imply that the Greeks of Herodotus' time (or of the time Herodotus was writing about) fought mingled together, nor does it imply that Herodotus thought the Greeks learnt to fight in formation from the Medes.
They often depict them fighting completely naked, too, and often show the Persians dressed up as Amazons rather than wearing actual Persian equipment. Greek artistic depictions aren't necessarily to be trusted.
Marching in step helps with fighting in orderly lines, but isn't necessary for it; the Romans didn't march in step, nor did most medieval infantry, but nor did they fight as disorganised mobs.
Is there any time when the occasion would seem to demand that he specify the depth of a formation? Because if not, the fact that he doesn't specify the depth doesn't tell us anything, really.
No they aren't. Herodotus says that the Spartans were each accompanied by seven Helots, but he doesn't say that these Helots fought mixed-in with the Spartans. (In fact, I don't think he says anything at all about how, or even whether, they fought -- they might have fought as separate units, they might have stood behind the front line and threw missiles, they might have just been unarmed camp followers. Any of these interpretations would be compatible with the text.) Indeed, he rather indicates that the Spartans fought in an orderly phalanx formation: in explaining the Persian defeat, he says that "in courage and in strength the Persians were not inferior to the others [i.e., the Spartans and Tegeans who were opposing them], but they were without defensive armour, and moreover they were unversed in war and unequal to their opponents in skill; and they would dart out one at a time or in groups of about ten together, some more and some less, and fall upon the Spartans and perish" (9.62.2) -- in implicit contrast to the Spartans, who presumably weren't without defensive armour and didn't dart around in small groups but kept in their formation. A little later he says that "[the Persians'] manner of dress, without defensive armour, was a very great cause of destruction to them, since in truth they were contending light-armed [γυμνῆτες] against hoplites [ὁπλίτας]" (9.63.2), again indicating that the Spartans were in general heavily-armed, not just a heterogeneous mob with only a small proportion of well-armed soldiers.
So wait, you believe one-off depictions on pots that you think support your conclusions, but not a general trend in the many artistic depictions of a lightened panoply? This isn't a case of heroic nudity; they're depicted fighting clothed and sometimes with helmets, but bronze and even linen cuirasses basically vanish from artistic depictions by the end of the fifth century. The Greeks considered the shield and spear the only essential elements of the panoply; when the Athenians implement their (brief) program of universal service in the late 4th century, the arms they issued at the end consisted of just a shield and spear. As I mentioned above, Xenophon's army of mercenaries, people who made their living fighting, had
very little body armor between them.
I didn't claim Herodotos's time was one of jumbled masses(Spartans potentially excepted, see below),, but that the preceding Archaic period was. Considering the fact that contemporary literary sources depict mixed battle lines and the archaeological evidence for extreme wealth stratification, the fact that Herodotos, living in a time of hoplite phalanxes, is even aware of mixed battle lines and thus makes a rather fanciful story to explain the appearence of the phalanx does seem to indicate it didn't exist from time immemorial. Moreover, the fact that Thucydides and Xenophon are both able to tell us how many ranks phalanxes fought in and Herodotos isn't is very suggestive; many an enthusiast has been frustrated by his account of Marathon, where he simply states the center was thinner than the wings without saying anything about the number of ranks. Sure, you can argue it's not strictly necessary, but that raises the question of why Thucydides and Xenophon bothered; if nothing else, it indicates a more lax attitude towards order in earlier times that ranks were considered unimporant. Taken in combination, the fact that Herodotos neither describes discernable ranks nor marching in step make it very unlikely that earlier periods, where we have literary depictions of mobs of mixed arms, were fighting in ordered phalanxes.
Regarding Plataea, you'll find Herodotos describes "The Lacedaemonians and Tegeans accordingly stood alone,
hoplites and light-armed together; there were of the Lacedaemonians fifty thousand and of the Tegeans, who had never been parted from the Lacedaemonians, three thousand." The emphasized passage would be redundant in context of the battle narrative if it wasn't describing their manner of fighting; we already know the Spartan contingent had many light troops, so there'd be no need to specify there were psiloi with them if he said the whole fifty thousand contingent was there. We know that in the time of Homer and Tyrtaios, armies fought with heavy and light armed together, so in that context, the survival of an archaic style mixed battle line makes sense. The hoplites naturally would have served as the promachoi, or front-fighters, but they were free to press forward or retire into the supporting mass of light troops for shelter in the Archaic period. Herodotos never uses the word phalanx in the military sense either, and never discusses Spartans marching in step, indicating that their military system hadn't yet fully evolved, so concluding the Spartans fought in an orderly phalanx (while being deployed together with their helots) requires a lot of interpretation compared to the view that they were promachoi.
A few key elements of Herodotos's narrative of the Persian invasion are contradictory, and we have to be careful about how we resolve them. First of all, we can be reasonably sure that, army vs army, the Greeks were not severely outnumbered at Plataea. Their army was massive, and a force much larger on the other side would be almost impossible to maintain in place for a prolonged period; moreover, the size of the Persian camp would roughly correspond to an army in the range of 60-120,000 men, and Mardonios never extends his battle line to envelop the Greek flanks, which would have produced an easy victory. Mardonios seemed to trust the quality of his troops more than their numbers; he wanted a battle before the Greek army could continue growing, and reportedly offered to fight with his Immortals against the Lakedaimonian hoplites with equal numbers (10,000 each); if he had a better chance of victory by dint of superior numbers, this challenge would have made no sense. These facts, derived from Herodotos's own narrative, tell us that the Greeks were not badly outnumbered overall.
As such, when Herodotos describes the actual fighting taking place between the Spartans and Tegeans against the Persians, we can be relatively certain the
Persians were outnumbered. Their contingent consisted of the Immortals and their armored cavalry, elite troops all. Earlier, in describing the manner of dress of the Iranian contingents of the army, he describes them all as wearing iron scale armor, and Masistius was so well armored the Athenians were at a loss for how to kill him for a short while. As such, his lines about the Persians losing because they don't wear armor don't make sense; pretty much nowhere else does he or any other Greek writer claim they defeated the Persians by dint of better armor. If anything, the battle of Plataea saw a comparatively well armored but badly outnumbered contingent stand their ground against overwhelming numbers until their leader was killed. This is based entirely on Herodotos's own account; when we incorporate archaeological evidence, the idea of well armored Greeks suffers still greater damage, since cuirasses are so rare compared to helmets among the Olympus panoplies. Of those who wore helmets, about a third seem to have had greaves, and a tenth cuirasses. This is of course excluding hoplites who wore no helmets, so the actual proportion of well armored Greeks is logically even smaller.
The concept of a large phalanx of well armored hoplites fighting in good order in the Archaic period really can't be sustained in light of the contrary evidence, which is why the new generation of scholars who are increasingly defining the consensus see the phalanx as a late invention, the product of changing patterns of land use at the end of the 6th century, rather than the product of the double grip shield.