There are several possibilities:
1) in ancient writings “Hebrew” and “Aramaic” are sometimes confused with one another.
2) Josephus might have been bragging.
3) Josephus was more polished in his Hebrew and Titus wanted to have the most able spokesman possible no matter what language was used.
4) Jerusalem was in the South. Aramaic might have been more common in the North where Jesus and the Apostles were from.
5) Even though only a few decades had passed, Hebrew might have experienced a resurgence in usage - which might partially explain Hebrew nationalist uprisings in the 60s.
6) Josephus might have been speaking only to an elite which knew Hebrew and little else or preferred to use Hebrew for their own particular reasons.
Was there ever a time when Hebrew was the mother tongue of the Hebrews? I wonder what the Moses generation spoke. Egyptian?
Or it was elites and invaders that spoke Aramaic and Greek, while the peasantry, who never had a reason to abandon it, still spoke Hebrew.
The Mishnah, which included records of debates from at least a generation or two after the destruction of the Temple, was composed in classical Hebrew. The generations that followed apparently had their discussions in Aramaic.
Several ancient sources attest to a gospel by the disciple Matthew as having been written in Aramaic, but with Hebrew letters.
Also, Aramaic is not the dead language people suppose. It persists today, among Syrian and Lebanese Christians. As I understand it, Hebrew is the language of the Jews (Judeans), whereas Aramaic was a language of the Gallileans and Samaritans. In the few places where Jesus’ language is retained in the gospels, it is Aramaic, yet Jesus certainly knew Hebrew as a rabbi who read scripture in the synagogue.
So if I understand you correctly, you are saying that Jesus spoke with a Drawl. So He actually said "Give us this day our daily bread Y'all."