Sushee – it is not so poshut! At least, Tosfos disagrees with you!
In the Gemorah in Kiddushin on 75b there is an argument among the Rishonim on this very issue. Some of the Rishonim actually present the opinion that a person is not Jewish if their father is a gentile and their mother is a Jew.
Tosofs says on Kiddushin 75b (DH Virebbe Yishmael):
Why do we need to say that Rebbe Elazar is holding like Rebbe Akiva that a non-Jew or a slave that has relations with a Jewess then the child is a mamzer? [Rebbe Elazar] can hold that in that situation the child is kosher [like Rebbe Yehoshua] and the problem is that we do not want a Jewess marrying a non-Jew. Why do we need the problem to be that we don;t want her to marry a non-Jew AND the child would be a mamzer (Isn’t one problem enough)?
Tosfos answers:
Perhaps we can answer and say that if [Rebbe Elazar] holds the child is kosher then he also has to hold that the child takes the status of the non-Jew, THE FATHER, and this is like that which we said before on 67b, the rules of who the child follows, and we don’t have to worry about a Jewess marrying a non-Jew [because the child is kosher]. Therefore, he holds the child is a mamzer and thereby the child’s status is being determined through the mother and we are also now concerned that the woman should not marry a non-Jew (Basically, you can’t have the problem of the woman marrying the non-Jew without having the problem of the child being a mamzer).
Before we take a step back and realize how amazing this idea is there is some clarification that is needed. Some of you might have noticed that this idea does not make sense. How could the child be KOSHER if he is going after the status of the father, the father is a NON-JEW? That would make the child a non-Jew not someone who is kosher.
The Maharsha commenting on this Tosfos (75b Virebbe Yishmael) answers this question by bringing in a Piskei Tosfos that clarifies Tosfos’ meaning. When Tosfos says that the child is kosher, he means that the child is a kosher non-Jew. If the child wants to convert to Judaism he will not be a mamzer. The Maharsha goes on to explain Tosfos as not holding like this as a second option and leaves Rashi (who says like Tosfos) as a Tzarich Iyun (needs looking into), but Rav Elyashiv in his Ha’aros on 75b tells us that Rashi and Tosfos do, in fact, hold that the child would be a non-Jew (that the child follows the father’s status).
So we see here it was not always so simple to say that if the mother was/is Jewish then the child is a Jew.