You can’t assert the NT as scripture/reportage on the basis that nobody was bothered to refute it - for a start, that would assume that people beyond a small sect considered it important at the time.
As far as specific beliefs go, you are correct. Can you prove that God spoke to Moses, or said do this and not that? But the history itself brings forth evidence. For example: If the Feast of Passover was an invention of man, how exactly did the leaders convince thousands of people to do it and pass it on in tradition through centuries especially since the claim was that it was based on a real historical event. If it was not based on a real historical event, I would think that someone somewhere would have said close to that time period, that’s not what happened and it never would have been able to even get started because most people would have known it was a lie from the beginning.
As for Christianity, the same could be said. Yes, the resurrection from the dead does require faith along with other recorded miracles. However, the question that rises is “why was he put to death?” “what were the charges?” Those events are historical and not faith based. If Jesus didn’t actually say the things he did, no one would have believed what was being said or written about it from the beginning because there would have been others that said, that isn’t what happened, that’s a lie. Some things do require faith but other things do not.
There is a difference between “Did Jesus really say _____?” and “Is ____ really true?” One is historical, the other is faith. The faith element was rejected by most people at that time, but the historical element is not. It is only in later times that people reject the historical elements as not being true, as they also do for the exodus out of Egypt and institution of Passover, but it is not rejected by the people who lived at that time and who were the most qualified to reject it as historical because they were there.
My point was on the historical elements that people can agree on should not be off limits to the discussion. The main question to answer from the OP is “How can we prove that Christianity is Judaism in its purest form and not a distortion of it”. I think you would find it extremely difficult to even say what Judaism is without using scriptures, or to say what is or is not a distortion of it. Unlike the other religions of that time period, Judaism and Christianity both claim to be based on real historical events, not some mythological fable. Removing the historical elements limits them to faith based and basically puts them in the same catagory as pagan mythology. So the answer to the OP is, you can’t.