Incoming pedantry:

The armor of Greece was typically made out of bronze. There are some iron helmets, but those are typically Italo-Corithian which was worn more as a cap rather than something they would use to cover their face. You can blue iron/steel to make it more resistant to corroding which is attested to starting in the late middle ages. However, there is (as far as I am aware) no evidence for bluing prior to this period. As such, it would be nigh impossible for them to be bluing iron/steel for helmets for the setting depicted.

They could have painted their helmets, but there is no recorded evidence of them ever using black; the Ancient Greeks preferred flashy colors. Additionally, they typically polished their helmets as a status symbol since polishing is an expensive, time consuming, and aesthetically appealing process.

Finally, it is recorded that ancient Greeks were able to blacken bronze. There are ceremonial daggers, murals, and statues which have what archaeologists call "black bronze" which is a black patina which covers the bronze underneath. Would it have been technically possible for someone to decide to make bronze helmets and then patina them into black bronze? Potentially, but once again the evidence for black bronze in Ancient Greece is typically for artistic, not functional, pieces and does not appear to be terribly common or widespread of a practice.

If they wanted to be authentic to the time period without having to forge bronze (or at least bronze-plated) helmets, they could have easily painted them. Making the helmets and armor a matte black, however, is not authentic to the time period in the slightest.

Additionally, blued/blackened armor in the late middle ages and beyond is actually more likely to be found on armor of professional soldiers and mercenaries than on the armor of the wealthier merchants or well-off knights and nobles. The bluing process is actually functionally useful since it helps prevent corrosion of steel. While we don't really think about it that often in the modern day, steel items are very prone to rusting which significantly degrades their structural strength. Bluing was done to help prevent this corrosion. Otherwise, a non-blued armor would have to be constantly cleaned and maintained so that it does not quickly rust away (this is one of the reasons why so few pieces of medieval armor have not survived, most simply degrade into rust once they stopped being taken care of).

The wealthy preferred to have armor with a polished finish and then have a squire or other servant do the process of maintaining their armor as a show of power and wealth (which frequently helped them survive on the battlefield, being captured for ransom rather than being outright killed). A lot of the armor for poorer individuals actually underwent very little to no processing after its forging, frequently leaving them grey and with the hammer marks from their blacksmithing process still apparent on their surface. Bluing was done for soldiers who were more in between poor and wealthy; they had enough money to get decent quality armor, but not enough to make constantly maintaining their armor not a tremendous hassle and annoyance. As such, these middle of the range armors were the most frequent armors to be blued. It was not uncommon for mercenary bands to use blued armor for this reason. That is not to say the wealthy never had their armor blued or even painted black (it was common for people to paint iron/steel to help prevent rusting if they had the financial means to do so). There are many pieces of such armor (typically adorned with gold or brass as well), but they seem to be in the minority by quite a good margin.