"It is the one to whom I will give this piece of bread when ..." (John 13:26)

"It is the one to whom I will give this piece of bread when I have dipped it in the dish." (John 13:26)
Here Jesus is answering a question from one of his beloved disciples as to whom would have Jesus arrested. Here is how it went:
One of them, the disciple whom Jesus loved, was reclining next to him. Simon Peter motioned to this disciple and said, "Ask him which one he means." Leaning back against Jesus, he asked him, "Lord, who is it?" (John 13:23-25)

So Jesus is answering this question. 

This means that Jesus' statement is in response to the questioning of his disciples, who were incensed that one of their own would turn Jesus in to the High Priest's guards for persecution.

This indicates that Jesus' disciples felt committed to their teacher.

It also indicates that Jesus knew precisely who was going to turn him over to the Romans. And that Jesus wanted his other disciples to know who it was. Why is this important?

Jesus was their teacher. He was trying to teach them something.

While readers of the New Testament have made so many of Jesus' statements to refer to some complex scheme, with Jesus being some kind of sacrificial lamb so that people over the next two thousand years could live selfish lives focused on wealth, fame and power; and be cleansed of their transgressions by Jesus' sacrifice.

No. Jesus' life was very practical. He was God's representative. He came to teach, and that is what he used his life to do. Jesus taught his students and disciples about God. He taught them to focus their lives upon God and serve God with all their hearts. Then he asked them to go out and teach this same message to others.
"He who receives you receives me, and he who receives me receives the One who sent me." (Matt. 10:40)
He told this to his disciples as he was sending them out to teach the same teachings that he had taught them. He was sending them out just as God sent Jesus. This makes Jesus God's messenger, and those Jesus was sending out his messengers.

And what message was so important for God to send Jesus to teach?
"'Love the Lord your God with all your heart and with all your soul and with all your mind.' This is the first and greatest commandment." ( matt. 22:37-38)