This is not really my area of expertise but it’s closely related to the things I do study…I’m guessing u/hannahstohelit or someone else could explain it better. But since it’s been a week and there are no other answers yet, I'll try to give the basics.

The Christian Old Testament is basically the same as the Tanakh - the Torah (Pentateuch), Prophets (Nevi’im), and history books (Ketuvim). The “rest of the Jewish texts” you’re talking about are Mishnah (collections of oral traditions about the Torah) and Gemara (commentaries on Mishnah texts), which are collectively known as the Talmud. There were different versions of the Talmud, the Jerusalem Talmud and Babylonian Talmud. Interpretations of the Tanakh are called Midrash. There were also studies of religious law, Halakha and Tosefta. (I apologize if I’ve missed any…)

The important thing for this question is that, aside from the Tanakh/Old Testament, none of these books existed yet when Christianity split off from Judaism.

At the time, Judaism was “Temple Judaism”. The Temple (actually the Second Temple) still existed in Jerusalem and priests who made animal sacrifices. There were synagogues and rabbis but they weren’t as important yet. In 66 (by the Christian calendar), the Jews rebelled against the Romans, and in 70 the Romans destroyed the Temple and expelled the Jews from Jerusalem. This was the beginning of "Rabbinic Judaism", where synagogues became the centre of Jewish communities and rabbis became community leaders. There were still Essenes, Karaites, Samaritans, and others who did not adopt Rabbinic Judaism, but the rabbinic form eventually became the “mainstream” in the Jewish communities elsewhere in the Roman world.

The Roman Empire eventually became officially Christian, and Jews generally weren’t allowed to hold any position of authority over Christians (as members of city councils, as judges, certainly not as slaveowners). They often weren’t allowed to build new synagogues or make renovations to old ones. Some Christian rulers were almost fanatically obsessed with the idea that Jews might take power somehow. Jews might be forced to live in their own separate neighbourhoods. They were only begrudgingly tolerated by the Christian majority.

Christian church doctrine developed the idea that the Jews were simply evidence of the superiority of Christianity, or proof that the Old Testament covenant had been replaced by the New Testament. Jews also had to exist to fulfill the conditions for Christ’s return (when they would all be converted), so theoretically, they should be left alone and the church should protect them. This didn’t always work out of course, and throughout the ancient and medieval periods, there were often forced conversions and attacks and massacres.

As far as Christians were concerned the Jews were almost an abstract idea, hardly even real people. They remained as described in the New Testament, fossilized in the 1st century. Christians didn’t know or care how Judaism developed after that.

I’m sure some curious people must have known about non-Tanakh texts earlier, but the church in general apparently only discovered them in 1239, from a French Jewish convert to Christianity named Nicholas Donin. The news reached all the way to Pope Gregory IX, who considered this a major scandal:

“If what is said about the Jews of France and of the other lands is true, no punishment would be sufficiently great or sufficiently worthy of their crime. For they, so we have heard, are not content with the Old Law which God gave to Moses in writing: they even ignore it completely, and affirm that God gave another Law which is called "Talmud," that is 'Teaching," handed down to Moses orally. Falsely they allege that it was implanted within their minds and, unwritten, was there preserved until certain men came, whom they call "Sages" and "Scribes," who, fearing that this Law may be lost from the minds of men through forgetfulness, reduced it to writing, and the volume of this by far exceeds the text of the Bible. In this is contained matter so abusive and so unspeakable that it arouses shame in those who mention it and horror in those who hear it.” (Grayzel, pg. 241)

The main problem was that the Talmud might (depending on how you interpret it) contain a few passages that criticize Jesus, so the church felt it might be heretical; really though, everyone was just shocked that the Jews continued to develop their own vibrant culture and society, instead of just remaining stuck in the 1st century as a sort of miserable, benighted people who had rejected the obvious Messiah. This discovery ultimately led to the “Disputation of Paris”, where Jewish and Christian leaders were supposed to debate the merits of the text. Naturally, it wasn’t much of a debate, and the church ended up burning thousands of copies of the Talmud.

So, short answer - the other books didn’t exist yet when Christianity split off from Judaism. And when Christians eventually discovered them, they weren’t very happy about them.

Sources:

Anna Sapir Abulafia, Christian-Jewish Relations, 1000-1300: Jews in the Service of Medieval Christendom (Routledge, 2011)

Solomon Grayzel, The Church and the Jews in the XIIIth Century (Dropsie College, 1933)

Hyam Maccoby, Judaism on Trial: Jewish-Christian Disputations in the Middle Ages (Farleigh Dickinson University Press, 1982)