Jewish Father and Non-Jewish Mother - Status of the Child

According to Halachah, a child is Jewish if his/her mother is Jewish, regardless of the father's status. The Reform and Conservative movements have challenged this (not so much challenge, as disregard) which unfortunately will cause many problems down the line. The arguments go back and forth, but I think it is interesting to note that challenging this idea is not new, and is already discussed and settled in the Midrash.

The Midrash Tanchumah, Parashas Chukas, Sif 6 states the following:

Yaakov from Kefar Giboraya [same guy who thought fish needed to be shcheted -ed] taught in Tyre that a child that is born as a result of the union between a Jewish man and a non-Jewish woman is circumcised on Shabbos. [Thus implying - in a strong way - that he is Jewish - ed]. Rabbi Chagai heard this and sent for him. He said to him, "How do you know this?" He said, "for it is written, 'And they were registered by family ancestry according to the house of their fathers.'" [He said to them,] "Prepare him for lashing." He said to him, "Is this the law that a person who says something from the Torah is given lashes?!" He said to him, "You do not teach correctly." He said to him, "How do you know?" He said to him, "Bend over [to receive your lashes] and then I will tell you." [.........] He said to him, "Based on what rabbi Yochanan said in the name of Rabbi Shimon bar Yochai. It is written, "And you shall not intermarry with them. You shall not give your daughter to his son, and you shall not take his daughter for your son." - Why? - "for he [the non-Jew] will cause your child [the son of the daughter with the non-Jew] to turn away after Me." Your son who comes from a Jewish mother is called "your son" but your son of a non-Jewish woman is not called "your son" but "her son," as it is stated, "but the son of the slave-woman as well I will make into a nation." He said to him, "Give your lashes because it is fitting and proper that I receive them."

We see here a biblical source for the law that Judaism goes through the mother. I have heard from other Rabbis (although I do not know their source) another interpretation of the verse. In this case the "he" in the explanation refers to the Jewish man, and the "child" is the Jewish man's offspring, which will be "turn[ed] away from Me", i.e. not Jewish. However, for the daughter to the non-Jewish man, this is not said.
It could be that I misheard this drashah, and I was told it in the manner of the Midrash originally.

Either way, the message is clear, that the Torah itself mentions (indirectly) the law. (Although in Yeshiva I learned that before Mattan-Torah, Judasim was passed down form the Father, and only afterward was it passed down through the Mother.)