Neo Deist
Th.D. & D.Div. h.c.
Let's look at your logic:
Pilate was cruel.
Pilate executed Jesus to make an example out of Him.
Pilate had to answer to Caesar.
STOP.
The above illustrates the nature of Pilate. It has nothing to do with Jesus. Back then, if you stood against Rome, you were eliminated.
CONTINUE.
Therefore Christians made Jesus a martyr.
They did so after the fact. Those that remained loyal to the cause wrote about Jesus decades later (Gospels) and changed certain facts/points about the story. The Book of Matthew goes out of its way to paint Jesus as the messiah, and to place the blame on the Jews.