Unlike most of religions, like Islam or Christianity, Jews don't take converts
Simply untrue. There have been extended periods in Israel's history where as many as 70% of immigrants were recent converts.
Making it hard to know at which point the Jews became actually a people.
They aren't a people.
Still, no doubt that a DNA census would be a good place to start with. But the outcome may make the conflict even worse if the results shows them as unrelated.
Conflicts can't be solved by DNA profiling. Nationalism is irrational.