So the NT should have NO connections at all with the OT then?
The New Testament stands on its own. The Old Testament is the religion of Judaism.
What about Jesus' own words in Matt 5:17-20?
I take your point but explicitly Jesus does not do away with the OT; rather he embraces it, particularly the Law. I think that rather than seeing the OT as purely Judaic it is more accurate to see the NT as the result of a progression in thought from old ideas (such as the erroneous belief that God kills as punishment) to the new - the unconditional love of Christ.
If you discount the OT you then throw out the Psalms, Proverbs, Ecclesiastes, Job - all speak to us down the centuries and are worth engaging with even as non-believers. Why ignore such richness of writing on the human condition and how we relate to God?
The words you quote from Matthew are particular to him, and it needs to be understood that Matthew was the Jewish writer, writing for a Jewish audience. He was concerned in presenting Jesus as a "good" Jew, and not a divisive teacher. These claims are not a feature of other Gospel writings. If we are to take the exhortation literally, then today Christians should be involved in animal sacrifices plus all the other numerous rituals including ceremonial washings etc etc, as recorded in books like Leviticus. Jesus said, "You have heard it said in times of old," or, "I bring you new wine for old." It is quite clear that Jesus was introducing a new and up-dated teaching. I am with Marcion who used the passage to justify a total separation between the religion that Jesus espoused and that of the Hebrew Scriptures.
I readily accept that there is much good teaching in Proverbs, the Psalms and Ecclesiastes, but they are not dependent on a belief in Judaism.