The Crucifixion did not "just happen" to Jesus: He *chose* it.
What Jesus chose -- the object on which He fixed His will -- was to do the will of Him Who sent Him. That the will of Him Who sent Him was that He be crucified was, if I may so put it, accidental to Jesus' choice.
I think we need to understand this logically prior choice to do the Father's will as distinct from the temporal choice to be crucified. Suffering as such is not good; suffering as such is lousy. Suffering sought out and offered to God like a dead mouse brought in by a housecat isn't much better.
"Son though He was, He learned obedience from what He suffered." Without fully addressing the question of how the Eternally Begotten Son can learn, this suggests that Jesus' human will underwent some temporal process of perfecting its obedience to the Divine will.
"Thy will be done," is the key, I think. Not, "I love You so much, I'm going to go suffer for You," but, "I love You so much, I will accept what suffering comes my way while doing Your will."
Do the cuts and bruises of everyday life constitute the crosses we are to take up to follow Christ? I would say they are a part of our crosses, even the sufferings we couldn't avoid if we wanted to. But a lesson of Gethsemane, I suggest, is that we can carry our crosses -- even our full crosses, with the persecutions in this world Jesus foretold for His disciples -- as followers of Christ yet in accordance with our own will; to be imitators of Christ, however, requires us to carry our crosses in accordance with the Father's will, and His will alone.
The above diagram shows the three ways in which our own wills can be conformed to the will of God. God's will is the blue circle. The red circle is the will of the prodigal, completely apart from God's will. The yellow circle is the will of the imperfect follower of Christ, lying partly but not wholly within God's will; this is the condition of the disciple who would go to Calvary without passing through Gethsemane. The green circle is the will of the perfect imitator of Christ, whose will is wholly determined by the will of God; there is nothing of self-will in what he wills.