"That's great, but it doesn't make you a Jew. "
What exactly in your mind is a Jew. Now according to the OLD testament there were 12 TRIBES. Only one was called Judah, the King line. We are told that the House of Israel was sent into captivity and eventually to the four corners and seared from their memory because of their sins of following other gods is the fact that they are the children of the House of Israel.
"Jew" comes from the Roman (Latin) term for Judean. Since that time it has become synonymous with Hebrew. A Jew is one who's mother is Jewish, or one who has converted to Judaism.
Someone who is born of a Jewish mother, or who has undergone a halakhic conversion.
Only one was called Judah, the King line.
To be technical, yes. But the name "Jew" has been used generically to refer to all identifiable "Hebrews" for a long, long time.
We are told that the House of Israel was sent into captivity and eventually to the four corners
Yes, but there were members of the 10 northern tribes who fled into Judah as well. So present-day Jews are really descended from all the original tribes, not just the tribe of Judah.
There have been a few cases of "lost tribes" who have sought to return to a connection with Judaism. Typically they have had their status "normalized" by conversion.
It's been gone over about 1000 times, but, hey, why not 30 more? Jews get to define who and what a Jew is. And that is this: someone born to a Jewish woman or someone who has undergone a conversion according to halachah. Let me point out that going to a seder does not make you Jewish.