It is popular to disconnect the plausibility of naturalistically acceptable claims made by the New Testament writers from those involving a supernatural commitment. This kind of an issue has come up before, when people have argued that no amount of accuracy on the part of the New Testament writers in recording mundane matter (such as we find in the Book of Acts), provides any evidence that the supernatural claims are also true.
It is interesting that some of you who are very eager to affirm the unity of the New Testament when you are arguing against the claim that the New Testament provides independent, or even partially independent strands of evidence for the claims of the New Testament, seem to want to argue that the New Testament is a diverse source when arguing that maybe we can accept the non-supernaturally involved claims while rejecting any element of the supernatural.
First, accurate reporting is a habit of mind. Even in mundane matters, you have people who take varying degrees of effort to get things right. If we conclude that a good deal of what our sources have to say is false, then that reflects poorly on everything they have to say, in much the way that evidence that a witness in a court case is untruthful about one thing can damage their credibility in other matters.
Admittedly, there are differences amongst the scenarios which conclude that the miracle claims are false. On some views, the writers experienced what they said they experienced, but were wrong about the causes of what they experienced. Hallucination theories of the Resurrection fit into this category. If the disciples hallucinated the risen Jesus, then they were "appeared to Jesus-ly," but were mistaken in supposing that the real risen Jesus was the cause of their being so appeared to. Or,t they might have seen lepers walking away from Jesus apparently cleansed, when the cause of this recovery was psychosomatic rather than divinely caused.
But other views treat the claims to be pretty much made up out of the whole cloth. If the New Testament contains a lot of material that was just made up, then it seems to me it would then be hard to credit passages that say Jesus taught that you should turn the other cheek.
I think the Gospels record actions on the part of Jesus that are mostly connected to his supernatural claims in one way or the other. Telling someone their sins are forgiven isn't directly supernatural, but if we accept it, we give ourselves the problem of figuring out how someone could believe that he had the prerogative to do so. Even Jesus's manner of teaching is a little odd from a naturalistic standpoint, in that you have a Jew who speaks on his own authority and even puts his own words (But I say unto you...)
in the place of the Law of Moses.
Now, you can argue, of course, that there are some good moral ideas that you can take from the New Testament even if Jesus didn't do or say much of anything he is supposed to have said. That's a different issue. I think something stronger can be claimed here; I believe that there is an ethical mind behind the Gospels that possessed true moral greatness, and that that is something that would have to be explained by any naturalistic theory. But that is a subject for another time.
But what I do maintain is that an easy separation between the naturalistic and the non-naturalistic is going to end up being a whole lot harder than it looks to carry off. It is indeed what the Trilemma argument is driving it.