That a lot of transitioning took place between the time of Josiah and the time of Jesus. A lot of perspectives had changed. You even see it in Jeremiah's writings and he was the SON of Hilkiah the Zadokite high priest; the fellow who was largely responsible for bringing in the revised history of Israel and the expanded "Law of Moses."
In the Pentateuch you see many variations where "God" is Elohim -(the English word "God" is always translated from "Elohim")- but then 'somehow' translates into "the Lord" or the "Lord God" (Jahveh). What I see in that is that -traditionally- the Children of Israel (COI), had worshiped Elohim (or the family of gods). The new Zadokite order attempted to change that into ONE god -Jahveh. So they wrote the text according to traditional tales BUT interjected their own Jahveh wherever they dared. (Remember that writing was a relatively new development which had only begun about 100 years prior to king Josiah. For hundreds of years before that, the tales were preserved by memory of especially assigned preservators to keep them "clean").
The COI believed in Abraham as their father. They wouldn't have believed in Elohim or the family of gods ..... as their father. They also believed in the law of Moses. But remember ....... Moses met the gods (Elohim or "God") on the mountain. He and the elders saw the gods eating and drinking on the mountain. The idea that he was up there alone with just Jahveh could and was -very likely- written in by the Zadokite college of priests/writers.
By the time Jesus came along, there may already have been a relatively large departure from Jahveh into another form of "one" and only "God." We can't tell too much because of the language change. (The New Testament was written in Greek whilst the OTestament was written in Hebrew). Yet, it seems to me that even from the high priests' perspective, they put very little emphasis on the name of their "God". I betcha they were calling him some variation of "Elohim" When Jesus cried out "Eli" on the cross (according to the text, anyway) it says that the people were asking if he was calling on Elias (Elijah). Well ........ the folks who watched him would have been familiar with the name of their God ..... so why would they wonder whom Jesus was calling to?