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It started after Christians decided to eliminate most of their Jewish heritage. Probably around 135 A.D. when the Bar Kochba revolt persecuted and killed those in the Christian sect for not participating in the revolt.
The name was clearly in all the OT Hebrew manuscripts and in the still-remaining scraps of the Septuagint from Jesus' time on earth. After the 135 A.D. revolution and the parting of the ways of Christians and Jews, the Christians began using (and copying, of course) the Septuagint. Since they apparently removed the "Hebrew" name of God from the Septuagint, they would surely do the same in the NT Scriptures also.
I've learned today that the early LXX mss still used God's Name, I didn't know that, thank you for bringing this up!
I've done some reading . . . it sounds like the LXX began to be copied with "Kurios" instead of YHWH following the practice of the New Testament writers quoting it that way.
I had thought that the NT writers did that because the LXX did so, but no, they did that as a new practice. And then they followed the same style with the LXX.
Much love!