The 3 kings is a western construct -- Eastern Orthodox and the Assyrian Church have numerous people and don't call them kings
Also, you are 100% correct that "Magi" or Zoroastrian priests would have come from the Parthian empire. They were most likely Iranis..
Nope. They were from what is now Turkey, most likely from around what is now SanliUrfa province, only a few miles from Haran, where Abraham sojourned with his father Terah and brother Nahor, until Terah died. Abram then left, but Nahor and his progeny stayed in this region, fro whence came Rebekah, Leah, and Rachel.
This region is called Anatolia to this day, not because it was east of Canaan/Israel, but because it was easterly of Macedonia, and therefore called in Koine Greek The Anatolia, and translated in our language "The East". Thus, what was called "The East" in the Hellenized Near East was actually direct north to northwest of Jerusalem and Bethlehem. Here are the phrases demonstrating what the Bible Greek says, from Matthew 2:1&2 --
ιδου μαγοι απο ανατολων
behold Magi from Antolias
(the two "Easts"--Anatolia proper between the Mediterranean Sea to the diagonal Taurus Mountains, and Southeastern Anatolia east of the Taurus range)
ειδομεν γαρ αυτου τον αστερα εν τη ανατολη
for we beheld the star of him in The Anatolia
======
The Magi did not come from Babylonia, IMHO. Think of the tableland of the Anatolias as the cradle of civilization proceeding from Noah and his sons. Think of the very, very ancient societies of Gobekli Tepe (click here), the observatory(?) of the astrologists in the area of Haran, from which Semitic/Aramaic religious tradition descend the Jews.