Just because Nazareth is not mentioned in the Old Testament, that doesn't mean it suddenly sprung into existence in intertestamental times or in the New Testament. There are many towns, cities and villages in Israel that are not mentioned in the Old Testament. Capernaum, Bethsaida, Bethany and Chorazin, for example, are all mentioned in the New Testament but not in the Old.The mistake was saying Jesus of Galilee . Nazareth was not a town, back in OT
As for it being a mistake (in your opinion) to say Jesus was of Galilee, I would just say that He is called "Jesus of Nazareth" seventeen times in Scripture. Was Peter mistaken when he said:
(Act 2:22) "Men of Israel, hear these words: Jesus of Nazareth, a Man attested by God to you by miracles, wonders, and signs which God did through Him in your midst, as you yourselves also know—