After rewriting this post a few times, I realized the main point I am trying to get at here is consistency. If you say that X is not important because it has property Y, then if I can show that Z also has property Y, then by your logic, Z must also not be important.
I saw this come up recently when an atheist said that there is a lot of violence in the bible. The Christian then replied that Jesus never advocated violence, implying that we have no biblical reason to be violent. I found this defense very interesting, especially since I have seen various versions of it come up quite a few times lately. What exactly is being said here? It sounds to me like they are saying if Jesus didn't say it, it's not important. If that is the case then we can skip all of the stuff that doesn't matter and save a bunch of time. (BTW, I would dispute that claim that Jesus didn't advocate violence, but that is not the point of this particular post)
Of course, this is not really what they are trying to say, I don't think they are trying to claim that only what Jesus said matters. I think they saw something in the bible that they didn't like and they are looking for an excuse to dismiss it. If something else comes up that they like from the Bible that Jesus did not say, they will probably not hesitate to use it. "It doesn't matter if Jesus said it or not, it's in the bible and it is good."
This is very similar to people who dismiss so many horrible things from the Bible because they are from the old testament. Those people will often turn right around and say how amazing the Bible is because of the ten commandments. The two ideas just don't make any sense together. Either the old testament is valuable or it isn't. Either the stuff not directly advocated by Jesus can be dismissed wholesale or it can't.