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The more I think the more makes sense to detach the Old Testament or at least treat it as closed.
Are Christians Jews? Jews has to circumcise their newborn due to the sign of their covenant with their god. But Christians (with a few exceptions that chiefly specific to Africa) have a new covenant, they don't circumcise their kids penises (for religious reasons). Ofc there are great million old Jewish traditions and laws Christians don't have to follow, simply because those are the things of the past, but circumcision is a very significant one.
In case of Jesus' life, his words and deeds, he was born into the Judaic religious landscape, he interacted with Jews ofc there will be mentions of the Law. But his death and resurrection closes all that.
Paul had to do his work largely in the Judaic crowd, he had to talk about stuff they recognized and understood.
But this is Christianity, using the already established knowledge of the surrounding people and incorporating it into Christianity, most of Christianity is baptized elements of pagan beliefs. The first such was Judaism, which actually a foreign element in Christianity.
The story of Jesus, his fate (death and resurrection, the circle of life, the travel of the light through the darkness, the triumph of light over darkness, the Sun's rout over the sky and underworld, the day cycle, the year cycle etc.), didn't originate in Judaism, but the religions of the region between the Nile and the Indus. The first ones who recognized him as God or the son of God weren't Jews, but people in the East, where the three Magi came from, at least one of them from the Kushan Empire (I wrote elsewhere about the Ethiop-Kus* mixup). Jesus wasn't the son of Yahweh, but another reincarnation, another avatar of Vishnu - at least one of the Magi knew him like that (the others knew him on other names for sure). He sold to the people as son of Yahweh, because in that area that was the one known, and in the grand scheme of things that detail didn't really matter, it could be sacrificed for the greater good.
But if he was Vishnu, why he was born among the Jews? Because you need the light the most, where there is the most darkness. The light born "from" the darkness and defeats it. And in that place Jesus had to address the Jews in a language they understood, he had to add honey for the bitter pill of the truth.
So Christianity to be Christianity doesn't need The Old Testament. In fact it doesn't really need most what's in "Christianity" (I put this into quotes since the beliefs of the denominations differ a bit from each other). Protestantism spilled quite a few things out of the bowl of Catholicism and still remained Christianity.