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 ends—the 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.
I do disagree with this point. I think Piper is falling into the mistake that most Reformed fall into of not recognizing who Jesus came and preached to and what He proclaimed. In addition to preaching salvation by Faith Alone in Christ Alone Jesus preached about the promised Messianic Kingdom. Jesus did not turn his attention to Gentiles until the Jews had rejected Him several times.
Piper like a lot of Reformed doesn't think much of Evangelical Christians. We agree with the Reformed on a lot of big points, the 5 Solas being the best example. However we recognize whom the audience was when the message was given and consider this significant, whereas most Reformed don't.
You bring up a good point. I think he was addressing what we see today with the “red letter only” theologies. Basically “Jesus did not say this or that, that was Paul.” I think he harmonizes this nicely for the audience.