If by “old law” one is referring to the laws of Judaism, as suggested by the thread topic, it is not true that “we” (meaning all mankind), would still be bound by these laws were it not for the redemption of Christ.
Jewish law was given by G-d to the Jews. The laws themselves consist of 613 commandments, but only a small subset of those - the Noachide Laws - are (were) binding on non-Jews.
According to Judaism, with the arrival of the Messiah, a new kingdom of G-d on Earth would come to pass, and most of the ills of mankind would vanish. The righteous dead of all faiths, not just Judaism, would rise up to everlasting life.
Christians believe Christ to be the promised Messiah, and that acceptance of His divine sacrifice creates in any one who accepts Him, redemption from the wages of sin and everlasting life beyond physical death. It is in that sense that He came to fulfill the old law.
I think for Christians the relevancy of Jewish Law is that it captured the teachings and commandments of G-d in the pre-Messianic times. Many of these teachings, at least the moral commandments, were reaffirmed by Christ. But it was never binding upon gentiles to follow all of the 613 commandments. G-d chose the Jews to preserve them through adherence and teaching.
Paul taught that acceptance of Christ was sufficient for all people - Jew and gentile alike - to secure G-d’s forgiveness and redemption. I don’t think this universal message is best captured by using phrases like “new covenant”, or “new chosen”.
The Jewish people still have their Torah, and to them G-d’s covenant is eternal. For Christians, Christ is the Messiah G-d promised the Jews (and everyone else), and Christ therefore fulfills Jewish law completely, from the Christian point of view.