The Ashkenazi have admitted that they arose from intermarriage of converts, which affects the gene pool. The question is whether they are the same people, and that is answered by the fact that peacefully merging into the same agreed polity as someone else makes you the same people. I've asked for proof that they are not the same people and none has come forth. In tenth century France, Ju was the short form and Judaeus was the long form of the same term for the same people.

If we Americans may tell Jews they're not true Jews and we know who are true Jews, Jews may tell us Americans that we're not true Americans and they know who are true Americans. Those who are othering Jews by calling them non-Jews are doing what they accuse Jews of doing.