In Galatians 4:4, Jesus was born under the Mosaic Law, so he was obligated to obey it, and he was sinless, so he set a perfect example of how to obey it, which included not breaking Deuteronomy 4:2, which forbids adding to or subtracting from the law. In Deuteronomy 13:4-5, the way that God instructed His people to determine that someone was a false prophet was if they taught against obeying the Mosaic Law, so if Jesus had done as you suggest, then those who reject him as being a false prophet for that reason would be acting in accordance with what God has instructed them to do. Jesus was one with the Father, so he was not in disagreement with Him about which laws we should follow, so he was transitioning his disciples away from obeying the Father. In John 14:15, Jesus used a parallel statement to equate his commands with those of the Father, and the alternative is to think that was hypocritically preaching something other than what he practiced. In John 14:24, Jesus said that his teachings were not his own, but that of the Father, which again leaves no room for him to be transitioning away from what the Father has taught.
About 1/3 of the verses in the NT contain quotes or allusions to the OT, so they certainly still considered it to be authoritative. In Romans 15:4, Paul said that OT Scripture was written for our instruction. In 1 Timothy 4:13, Paul encouraged him to devote himself to the public reading of Scripture, which is referring to OT Scripture. In 2 Timothy 3:15-17, Paul referred to holy writings that Timothy had available to him since childhood, which could only be referring to OT Scripture because none of the books of the NT had yet been written at that point, and he said that it as profitable for teaching, reproof, correction, and training in righteousness that the man of God may be complete, thoroughly equipped to do every good work. In Acts 17:11, the Bereans were praised because they diligently tested everything that Paul said against OT Scripture to see if what he said was true, so Paul should not be interpreted as saying things that they would have outright rejected as being the word of a false prophet. In Jeremiah 31:33, the New Covenant still involves following the Torah, which is also known as the Mosaic Law.