Challenging Jewish matrilineal descent

I am a Jewish man married to a non-Jewish woman, and resent the fact that Orthodox and Conservative Judaism would require my offspring to be converted in order to be considered Jewish. As far as I'm concerned, if my children feel Jewish, they are Jewish. In actuality, there is no biblical basis for the matrilineal dictate. In biblical times many Jewish men married non-Jews, and the religion of their children was passed by the father. Judah, Joseph, Moses, David, and Solomon all married non-Jewish women, and their children were obviously accepted as Jews. In addition, passing of the status of Kohen continues to be patrilineal. The rabbinical switch to matrilineal identification occurred around 70 AD, around the destruction of the second great Jewish temple, and may have been intended to identify the children of raped Jewish women as Jewish. However, the rabbinical justification of matrilineality is based on a weak and convoluted interpretation of a bible passage banning intermarriage. Fortunately, both Reformed and Reconstructionist Judaism have come to accept patrilineal as well as matrilineal descent as a basis for Jewish identity and even accept as Jews adopted children raised to be Jewish, even when the religion of the children's parents is non-Jewish or unknown. The matrilineal rule is one factor that has kept the Jewish people small in number and weak. As an American, I have enjoyed growing up with a feeling of strength as part of my identity. I find no joy in feeling weak. I feel that if the Jewish people are to survive and thrive, it must make some changes, a major one being abandonment of the antiquated rule of matrilineal identification. It is unfortunate that Israel uses that rule because its small religious minority elects parliamentarians who form a swing vote almost always needed to form a government. Therefore, Orthodox rabbis have inordinate power in Israel over identifying who is considered a Jew. I would like to see the authority over Jewish issues in Israel shared over all the branches of organized Judaism, which would allow people to be accepted as Jewish according to the Reformed and Reconstructionist interpretations.