“Based on DNA, most of your ancestry is probably from converts (at least on the female side).”
Not true. I had my DNA checked and I match the DNA of most Middle Easterners – Specifically the strain that originated between Iran and Iraq (Ur Kasdim), even though my family has been in Europe for over a thousand years. Being a Kohein, I am pretty sure that none of my forefathers married converts.
Most Jews share this same DNA – specifically “J2.” I am J1 like most Kohanim, Sephardi and Ashkenazi.
We are called Jews because we follow our fathers’ tribe. Even if every generation married kosher converts they would still be Yehudim, so I don’t see your point. Those from Binyamin and other stragglers would still have lived under the rule of Judah. As for me, I tell most people that I am not a Jew… I am a Kohain (just to get a reaction 😉 )