Yerusalyim....Yes, you are absolutely right. I wish the film paid more attention to historical accuracy than Christian tradition....but after all, this is just a passion play....
WillPower....There is absolutely no evidence that the Romans had prisoners carry a pole to their execution. As Yerusalyim said, classical Greek and Latin sources show that Roman executioners forced prisoners to carry the crosspiece. The crosspiece in Latin was called the patibulum, and the practice of making the prisoner bear it existed even before crucifixion came into existence. When the Romans borrowed crucifixion as a method and technology of execution from the Phoenicians during the Punic Wars (who in turn had borrowed it from the Persians), they combined it with their older patibulum-bearing practice to produce the ritual that Jesus was said to have gone through. The person who would bear the cross would stretch out his hands and have the crosspiece either tied across his back or under his chest.
Dionysius of Halicarnassus (first century B.C.) for instance says that "the men were ordered to lead the slave to his punishment, having stretched out both hands and fastened them to a piece of wood which extended across his breast and shoulders as far as his wrists, followed him, tearing his naked body with whips." Plautus (second century B.C.) described crucifixion as spreading out the hands and nailing them to the patibulum, and twice refers to the ritual of bearing the patibulum before crucifixion: "I shall bear the patibulum through the city; afterwards I shall be nailed to the cross....I bet the hangman will make you look like a human sieve, the way they'll prod you full of holes as they run you down the streets with your arms on a patibulum." Plutarch in his Moralia similarly said: "Every criminal that goes to execution must carry his own cross [stauros] on his back." Since there was no separate word in Greek for patibulum, the word stauros could refer either to the single crossbeam or the entire cross. Note that Plutarch describes the cross as on the criminal's back. John 21:18-19, which predicts Peter's martyrdom, also has Peter stretching out his hands before he is led off to his death. The traditional picture of Jesus carrying the whole cross over one of his shoulders at an angle is not consistent with the historical practice, which involved stretching out the hands and carrying the cross over the back or chest. This was done often because the stipes (vertical pole) was often kept stationary in place, and the criminal bound to the patibulum would then just be lifted up to the stipes after bearing it through the city.
As far as I know, the earliest artistic depiction of Jesus carrying the cross over his shoulder was in the fifth century, long after crucifixion ended as a form of capital punishment.