Belief that Jesus existed and even doing good works in his name doesn't make a person a Christian, or make themcovered by the new covenant.
Jesus answered and said to him, “Most assuredly, I say to you, unless one is born again, he cannot see the kingdom of God.”
4 Nicodemus said to Him, “How can a man be born when he is old? Can he enter a second time into his mother’s womb and be born?”
Interestingly, Perry, this example you used to prove your point is one of the passages we know that Jesus never would have said. Why, you may ask?
It's a pun! More specifically, it's a pun that only makes sense in Greek. In Aramaic or Hebrew, Nicodemus never would have confused the words for spiritual birth and physical birth, they are totally different. But in Greek, however, the words were the same.
This passage, to make sense, could ONLY have been spoken in Greek, not any language two Jews would have been speaking to each other at that time and place with their backgrounds.