You are absolutely right. John does say that Jesus carried his own cross without help, which is entirely consistent with John’s higher Christology as compared with the synoptic gospels, especially Mark:

Mark:
Doesn’t know who touched him in the crowd
Prayer in the garden: “Remove this cup from me.”
Carrying the cross: Simon of Cyrene
Jesus’ last words: “My God, my God, why have you forsaken me?”

John:
Knows the entire life history of the Samaritan woman
Prayer in the garden: that the disciples not be scattered
Carrying the cross: himself
Jesus’ last words: “It is finished.”

Further, Jesus in John’s gospel is never presented as anything other than in total control: “I lay my life down of my own accord, no one takes it from me.”

What these accounts make clear is that these are theological tellings rather than historical. It is interesting to see the intersection of faith and history (as in the original story about Alexander and Rufus), but at the end of the day, the lessons of the gospel narratives are not dependent on that.