Oh this is a good question, especially because I don't think I've ever read someone who commented on the fact that this particular modern trend did not hold true for Cyrus the Younger's mercenaries (or other Greek soldiers in Persian service). I'm not personally able to comment in detail on ancient mercenaries in other eras (a la the Roman Foederati), but Greco-Persian mercenary relations is more than enough to take up a long discussion. So I'll stick to the late-5th and early-4th Centuries BCE around the Aegean Sea.

The pay rate of the regular military was dependent on the fortunes of the state's fortunes. According to Thucydides, at the height of Athenian power during the Peloponnesian War, Athenians hoplites and their footmen/servants on campaign received 1 silver drachma per day. Other references suggest that this was pretty typical for soldiers in prosperous city-states overall, though nowhere is as clearly documented as Athens. However, this is at least a good baseline to work with since the most famous individuals in The 10,000 include Xenophon, an Athenian veteran of the Peloponnesian War, and several Spartans whose city was at its peak during Cyrus' revolt.

All of the documented mercenary pay in late-Classical Greek sources is less than the 1 drachma rate, with the possible exception of a reference from Demosthenes to Phocis desperately offering double "the usual rate," in the 350s. The usual daily rate, depending on context was anywhere between 2-4 obols, valued at 1/6 of a drachma each. 2 was considered a bare subsistence wage. So double in Phocis could be anywhere from the high end of someone else's usual rate to slightly more than the Athenian army's pay at its height. However, 4 obols/day seems to have been most common by the mid-4th Century BCE, when several Greek comedies referred to mercenaries as "Four Obol Men."

Cyrus the Younger himself actually played an important part in setting that standard too. His rebellion was not his first brush with Greek soldiers. At the tail end of the Peloponnesian War, Cyrus was sent to western Anatolia to oversee Persian support for Sparta against Athens in 407 BCE. The Perso-Spartan alliance struggled against Athens for most of the preceding decade, routinely hampered by disputes over wages and the local Persian satraps fluctuating between actually aiding Sparta and trying to play the two Greek powers against one another to weaken them both. Part of the treaty between the Spartans and Persians specified that those Satraps would pay for the wages of the Spartan soldiers and sailors at a base rate of 3 obols/day. One of Cyrus the Younger's first acts upon taking command (at just 16 years old) was to raise that to 4 obols/day and make arrangements to ensure payment was always delivered on time. A few years later when Cyrus started recruiting mercenaries for his campaign against Artaxerxes II, the new recruits expected and received the same rate.

There are a few factors that explain why these ancient mercenaries served for less than what they would expect at home.

One is simply supply and demand. The decades-long Peloponnesian War in the late-5th Century left the wider Greek world with a glut of experienced veterans with no war left to fight at home just as things started heating up across the western Persian Empire, southern Thrace, and Sicily. This was further enhanced by a general economic downturn playing out in Greece. An influx of foreign grain mixed with the devastation wrought by recent warfare to render southern Greek farmers uncompetitive in the market, ultimately leading many former citizen-soldiers to lose their land and seek alternative employment.

In the more economically prosperous classes and city-states, military experience was also an important tool to climb the social ladder. Veteran commanders were more likely to be elected to high office and the responsibilities of military leadership in the future. Ambitious men who came of age after the Peloponnesian War or failed to distinguish themselves in that conflict went out to seek notoriety in foreign armies and bring those credentials home.

Financially, ancient mercenaries also had an incentive that their modern counterparts do not (legally at least): loot and pillage. It was standard practice for armies to pillage the camps or cities of defeated enemies and steal valuables for themselves. The more grandiose the target, the higher the possibility for plunder, especially in punitive campaigns against rebels or invaders. This is a recurring theme in Xenophon's Anabasis as The 10,000 repeatedly pillage territory they passed though and had to abandon their cumbersome loot.

Finally, one of the biggest contrasts between these ancient mercenaries and modern mercenaries is that the soldiers of fortune today tend to be specialists of some sort. Maybe they're genuine special operators or experts, maybe veterans from a more developed and complex military, or maybe they are simply veterans in a new war zone. In general, none of that was true for Greek mercenaries hired by Persian leaders. European Greek hoplites and peltasts weren't any more elite than their cousins in Ionia, Lydian and Phoenician heavy infantry, or light infantry from the Caucasus and Zagros mountains. Persia was the more developed and complex power. Many of the local levies in the 4th Century Persian Empire were just as likely to be veterans, sometimes of the same wars.

There simply wasn't anything "elite" about the mercenaries hired by Cyrus the Younger or other Persian governors and rebels. They were just there to add more bodies to the army and make the local governors less dependent on Empire-wide drafts (and/or capable of revolt). The 10,000 (actually about 14,000 when Cyrus started his march toward Babylon) were just Greeks soldiers collected from a wide variety of backgrounds. Anything elite about them mostly developed as a consequence of their experiences in Persian territory, learning how to effectively survive and fight Persian armies in their home-territory over the course of months. That certainly made them a valuable asset in 399 BCE when the remains of The 10,000 (down to about 8,000 by then) were hired by Sparta to join their invasion of Persian territory, but they weren't special from the start.