No.
Judah, Benjamin and Levy survived more-or-less intact. Shimon’s portion was mostly within Judah, so it is safe to think they were included with Judah. These were joined by remnant faithful of the other tribes when the two kingdoms split.
The DNA profiles of all the tribes would be similar, and related most closely to others from the Mideast.
“Jews are descendants of Judah and not of all Israel.”
Grammatically, yes. In reality, we’re all mixed up.
I’m a Levite, theoretically. But I certainly call myself a Jew.
The Jews of Jesus' day were NOT just descendants of Judah. Benjamin, Levi and varying amounts from other tribes too had, after the Assyrian invasion of the North, made it back into the South of Palestine (the region of Judah).
The idea that the ten tribes in the North were "lost" is a romantic 19th Century invention. The ten northern tribes--after centuries of idolatry--and intermarriage-with-pagan-tribes (those things went together) got conquered by the Asyrians, who had a policy of displacement of peoples...and resettlement of their own people there. Hence the Northern tribes became intermarried, conquered, displaced--and totally diluted--into the (despised) religiously and ethnically diverse Samaritans by Jesus' day.
While the South too was conquered--and moved to Babylon--their idolatry wasn't as total--Babylonians didn't displace them in Palestine--and they maintained their ethnic and religious identity in Babylon (and later Persia)and 70 years later returned to the land.
No similar thing happened with the Northern tribes (they did NOT maintain their religious/ethnic identity...and they never, en mass, returned to the land.)
So while modern Jews can be said to be primarily from Judah, Benjamin and Levi (the priestly tribe) figure in there as well, at least.