I'm similarly perplexed by people who accept the 'red letter' sayings of Jesus in the gospels, but do not consider the same writers & documents to be reliable in other respects.
When conversing with "red letter only" adherents, I point out that they should also take into consideration not only the Words of Christ but the actions. Some fail to see the Gospels in light of the cross and empty tomb. John Piper puts it this way:
Every verse of all four Gospels is meant by the authors to be read in the shadow of the cross. When we start reading one of the Gospels, we already know how it endsthe death and resurrection of Jesus as a substitute for our sins (Mark 10:45; Matthew 26:28)and we should have that ending in mind with every verse that we read. And this is exactly what each of the Gospels intends.
The sermon can be found at Did Jesus Preach the Gospel of Evangelicalism?
You can view or read the sermon. Excellent piece when you get the time.