Is Matrilineal Jewishness Valid? The Answer Is...

Yes, but not because of the P'rushim is Matrilineal Jewishness valid. I should've at least saved my posts from one group that I was in with Asher Romero ben Israel and Adam Klein (who is not Nehemia Gordon's friend, as he stated, and clearly not my friend-- on his own volition), because I'm so damned lazy and don't really want to rewrite what I already wrote (Maybe I'm more depressed lately as well, thus more lazy. Anyway...).

First, let's begin with B'resheet 17:19-21:

¹⁹ Then God said: “No, Sarah your wife shall bear you a son, and you shall call his name Isaac; I will establish My covenant with him for an everlasting covenant, and with his descendants after him.²⁰ And as for Ishmael, I have heard you. Behold, I have blessed him, and will make him fruitful, and will multiply him exceedingly. He shall beget twelve princes, and I will make him a great nation.²¹ But My covenant I will establish with Isaac, whom Sarah shall bear to you at this set time next year.”

We see that Yitzhak had to have Sarah-- not anyone else-- as his mother and Avraham-- not anyone else-- as his dad (which is why, incidentally, I believe that some Kara'im argue that both parents must be Jewish in order for the child to be Jewish). We also see that Yitzhak was the first Yehudi:

²² For it is written that Abraham had two sons: the one by a bondwoman, the other by a freewoman. ²³ But he who was of the bondwoman was born according to the flesh, and he of the freewoman through promise, ²⁴ which things are symbolic. For these are the^[d] two covenants: the one from Mount Sinai which gives birth to bondage, which is Hagar— ²⁵ for this Hagar is Mount Sinai in Arabia, and corresponds to Jerusalem which now is, and is in bondage with her children— ²⁶ but the Jerusalem above is free, which is the mother of us all. ²⁷ For it is written:

“Rejoice, O barren,
You who do not bear!
Break forth and shout,
You who are not in labor!
For the desolate has many more children

Than she who has a husband.”^[e].

Yitzhak was "born of the promise". We see that in this way, Yitzhak was the first Jew and that Patrilineal Jewishness is as valid as Matrilineal Jewishness. But let's get back to Matrilineal Jewishness. We've seen so far that the contribution of the mother cannot be left out (thus part of why I stated that  I believe that some Kara'im argue that both parents must be Jewish in order for the child to be Jewish). The contribution certainly was not left out in the cases of the young Dani of Mitzrayimi parentage, Hiram the king of Phoenicia and Huram his dad(? See Wikipedia for the dispute on this.), and Timothy HaYavani:

• ^"10 Now the son of an Israelite woman, whose father was an Egyptian, went out among the children of Israel; and this Israelite woman’s son and a man of Israel fought each other in the camp. ¹¹ And the Israelite woman’s son blasphemed the name of the Lord and cursed; and so they brought him to Moses. (His mother’s name was Shelomith the daughter of Dibri, of the tribe of Dan.) ¹² Then they put him in custody, that the mind of the Lord might be shown to them. ¹³ And the Lord spoke to Moses, saying, ¹⁴ “Take outside the camp him who has cursed; then let all who heard him lay their hands on his head, and let all the congregation stone him." The P'rushim use this, but only to attempt to invalidate Patrilineal Jewishness; thus, they don't use this for the right reasons. 
• "So he said, “What kind of cities are these which you have given me, my brother?” And he called them the land of Cabul,^[a] as they are to this day." Here, Hiram calls Shlomo "brother" because " ³ Now King Solomon sent and brought Huram^[a] from Tyre. ¹⁴ He was the son of a widow from the tribe of Naphtali, and his father was a man of Tyre, a bronze worker; he was filled with wisdom and understanding and skill in working with all kinds of bronze work. So he came to King Solomon and did all his work. " Hiram also blessed G-d. "Blessed be the Lord God of Israel, who made heaven and earth, for He has given King David a wise son, endowed with prudence and understanding, who will build a temple for the Lord and a royal house for himself!"
• Then he came to Derbe and Lystra. And behold, a certain disciple was there, named Timothy, theson of a certain Jewish woman who believed, but his father was Greek. ² He was well spoken of by the brethren who were at Lystra and Iconium. ³ Paul wanted to have him go on with him. And he took himand circumcised him because of the Jews who were in that region, for they all knew that his father was Greek. ⁴ And as they went through the cities, they delivered to them the decrees to keep, which were determined by the apostles and elders at Jerusalem.

One more quick example; which is not Scriptural, nonetheless valid: Geraldo Rivera's daughter Sol. Are we to say that she is not Jewish because her dad's paternal granddad was a Spaniard and his paternal grandmother a Taino Native American? Is Sol then only a Spaniard of Taino and Jewish descent? To rule out Lillian Friedman Rivera's contribution and Erica Levy Rivera's contribution (which, by the way, makes Sol a Matrilineal Levit) would be entirely unfair and unpractical, especially since Sol is 3/4 Jewish.