While you have gone to great length to attempt to prove that "British-Israelism" has its roots in the so-called "Lost Tribes" of Israel theory, the substance of such is lacking, as proof, for so many reasons. I want you to know that I have read your post in its entirety, most of the material of which is commonly found on those "British-Israelism" or "Anglo-Israelism" web sites.
I'll have more to share on how "British-Israelism" has reached even into the highest realms of the British Monarchy in our lifetime, or at least in my lifetime. I'll share the many different ways that "British-Israelism" fails to offer the Christian world any hope whatsoever.
While this theory has prevailed for more than three centuries, it hasn't been until the past couple years or so that the theory has been invalidated and dismissed as of no substance. This evidence would have made my academic work so much easier had I known all this when I was working on a MA in Christian Apologetics, I had to do extensive research on the whole issue of Anglo-Israelism.
The evidence that refutes "British-Israelism" or "Anglo-Israelism" is one of my favorite interests, that being the study of DNA. DNA proves that the majority of the people of the United States and British Commonwealth did NOT have the same patrilineal roots as the Jews, from either tribes of Israel or Judah. The issue is not one of race, but it is a fact that the Jews and the peoples of the USA and the British Commonwealth did not have the same father as an ancestor.
Those who hold that the people of the USA and the Brits are descendants of the so-called "lost tribes of Israel" have been proven totally wrong by the science of DNA: the very evidence that would be proof in a modern day court of law, to prove paternity.
Since the conclusion of the Human Genome Project, human DNA has been mapped thoroughly: and the functions of the genes have been identified and the associations of the mitochondrial chromosome and the Y-Chromosome have been traced. The Y-Chromosome has proved to be exceptionally stable over thousands of years. The genetics of the fathers have been passed to the sons with so little variation that familial relationship can be determined with great confidence.
Specifically, Scripture says that the tribes of Israel originally came from Abraham, as did the Arabs. DNA shows clearly that the Jews and Arabs of today have the same patrilineal ancestry, but the Western Europeans do not, and that includes the UK countries.
There are those who base their entire religion on "British-Israelism" or "Anglo-Israelism". Many of those intensely believe that "British-Israelism" is The Key to Prophecy. They have their entire religious prognostications based on Scriptures in the Old Testament of the Bible based on the events that were to unfold upon ancient tribes by claiming that statements pronounced on those people apply to modern nations in the 21st century. These religious sects/cults (Worldwide Church of God) and its offshoots, used to predict these things would transpire in the 20th century, but the events predicted never happened. We have recorded, nearly 300 prophecies of Herbert W. Armstrong, of which not a single one came to pass. Thus Herbert W. Armstrong was a "False Prophet" in every respect. There is no other possible outcome: The basis of their religion is false. They are wrong, They will always be wrong. There is no possible future in which they will ever be right on this matter. None of the things they tell their followers is true.
The likes of Arnold Murry of Shepherd's Chapel, the LDS Church teachings, SDA's, Christian Identity Groups, and even Jimmy Swaggart, and so many that I have named in the breakup splinter groups of the former Worldwide Church of God, have all incorporated "British-Israelism" into their teachings in one way or another. They certainly know by now that they are in error, yet they still preach empty promises and entice people into their religious groups with an empty eschatology. Many people are deceived and drawn into their false religions because world events are happening at such a breakneck pace and they fear that the religions espousing British-Israelism will save them from the Great Tribulation that lies just ahead.
A book that was published just last year has shown that 90% of the Jews in Israel today are not "Biblical" Jews at all, but have genetic DNA that connects them with the "Khazars" a tribe that once occupied the region of central Eurasia. In Israel today, the two major groups are either "Sephardic" or "Ashkenazi", the "Ashkenazi" are tracked back to the "Khazars" in the Step Plateau of central Eurasia. The "Sephardic" Jews can track their Biblical identity back to Abraham, Issac and Jacob.
This global DNA study has become a problem for the folks in Israel today, and one can understand why they do not want to endorse the DNA study findings. They actually are stonewalling the findings of the world geneticists involved in the Genome Project.