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.
I take your point about nationalism being irrational but you can break it down by dispelling the myths that surround it.
showing the people that they are related goes a long way to dispelling the myth that they are foreigners.
it wont fix the problem but it will be one more step along the road to reconciliation.
I agree for almost all of the last 3000 years Jews have not been one people. the kingdom of Isreal briefly united the Jews for 100 years or so back in 1000 BC and that's about it.
just like Christians have never really been one people.