But if I my wife and I were to convert, then any children born after would have a Jewish mother. I don't understand why genetics has to enter into it?
Kohen a part of the Levi tribe who are the "priests" who would do the sacrifices in the Temples are not aloud to marry converts, and like all Jews are not aloud to intermarry at all.
You completely missed my point.
As I understand it, the question at hand is: Who is Jewish?
Answer: Anyone with a Jewish mother and who keeps the Torah.
Scenario: Though not born Jewish, one's parents convert. They go through Bet Din. They practice Torah. They keep the Talmud. Their children are born after their converstion. And they raise their children Jewish.
I hear you talking about the importance of being able to trace your genetic code back to prove that you were Jewish.
My particular question is: Why that would be necessary? Wouldn't the children of this couple be as Jewish as any Kohen Jew?
And here is why I think the answer to that question should be: "Yes, these children are Jewish. It isn't necessary to trace your genetic code back like the Kohen can in order to prove that you are Jewish."
Because, as has already been said above. If your mother is Jewish and you were raised Jewish, and you keep Torah, then you are Jewish. It shouldn't make any difference what your grandmother was.
Just to provide some back-up to my personal opinion, let me give an answer to a similar question I found elsewhere (this on a Jewish website):
Question
Very basically, my mother converted to Judaism for the wrong intentions, to marry my father. The conversion was done, to my knowledge with the proper guidelines of an orthodox conversion, in front of a Bais Din and Mikveh. I need to know if am I really Jewish? And if not, what do I have to do to become Jewish?
Answer
The Bet Din usually refrains from converting people who have ulterior motives such as marriage - (source: "Code of Jewish Law" Y.D. 268:12). However, once a person is converted by the Bet Din, and there was a serious undertaking to study and observe the Torah, the conversion is valid.
So, I don't understand how having Kohen DNA or not, has anything at all to do with one being Jewish?