Wasn't the "much greater area" ruled by Herod the Great called, simply, Judea (it would not have been spelled ioudia in Greek, would it?), and split up into Judaea, Galilee/Perea, and Iturea/Trachonitis only after Herod the Great's death? In which case, Luke is quite right in calling Herod the Great Herod King of Judaea.
No. Herod the Great's title was technically King of the Jews (granted by Rome in 40 BC according to Josephus and 39 according to Appian), not King of Judaea. This is a subtle, but important difference, like Louis Phillipe being 'King of the French' instead of the Bourbon 'of France'.
We see this referenced in Matthew when the Magi ask about a new King of the Jews and again when Jesus is crucified. Pilate writes it on the titulus over the cross, thus a Roman grants the title to Jesus as well, a nice juxtaposition to the secular reign of Herod the Great, but I digress.
Regardless, to Romans and Greeks the whole levant was loosely termed Syria. The later province of Judaea was in fact a subdivision of Roman Syria, hence the reference to Quirinius its governor. Judaea to Jewish individuals refers to the old Juda, the southern Kingdom of the OT. To the Romans Judaea meant very little, like if I reference the Limpopo province of South Africa, then most will likely have little idea what I am talking about. First century writers would mention Herod's Imperium (ie of Jews) as his identifier and if referencing his territory they would likely list the various areas like Judaea, Samaria, Idumaea, Galilee etc. Often Imperium and a territory were co-terminous, but not in Herod the Great's case. A good example of what I am talking about is when 'provinces' were granted to proconsuls to wage a specific war, like when Sulla was given the provincia/imperium to fight the Mithridatic War or Pompey to fight mediterranean piracy.
We see in the gospel that a lot of effort is made to differentiate Galileans from Judaeans and we see Idumaeans, Samaritans and so forth clearly identified as such in the texts of the first century. We tend to see titles as referencing territory, but the Roman concept of Imperium is quite different. It references a sphere of influence which extends beyond territory to individuals as well, hence Pilate sends Jesus to Herod Antipas as he falls in his Imperium. Often Imperium overlapped other's territory as when Pompey was granted Imperium over the mediterranean coast without the provincial governors losing sway here either.
Herod the Great held Imperium over all the Jews, thus the Romans expected him to keep order wherever Jews were found, whether it was within his territory or not.
While it is possible that Luke could reference Herod as King of Judaea in a similar manner as people call Elizabeth II the Queen of England instead of the UK, this seems out of character to the otherwise careful and precise history Luke had written. As you mentioned, he lists multiple rulers and gives precise information to fully place it in the right context. Such sloppy titulature from Luke seems unlikely.
Also Luke is careful to list a number of people in power simultaneously in 3:1-2: Would he have been content to only mention Herod Archelaus in Chap 1, when Antipas and Philip also ruled?
The question can equally be asked why he didn't mention multiple people then as well? He later mentions Tiberius, the High Priest, the Governor etc. in chapter 3.
The fact is Luke does even if a few less than in chapter 3: He mentions Augustus, Quirinius and Herod. Chapter 1 is dated sufficiently by the references in Chapter 2 as both are set a few months apart. Quirinius and his census though suggests the Herod here is Archelaus.
The difference is that Chapter 3 sets the epoch based on various methods of dating for the various areas in which the gospel is set. It is necessary to mention the whole list to securely establish roughly when it occurred as it is a much longer period.
The first two chapters are only set in Judaea, so referencing its ruler, the Emperor, the Syrian governor and an important event, the first Roman census, should be more than sufficient to date it.
