I purchased the old lens as you and tried to fix myself as my winter hobby :-). The autofocus issue can vary from lens to lens. The most possible reasons could be:
1. the broken flex-cable.
2. the old not function well USM motor.
3. the dirty status ring inside to tell the position of the focus ring.
4. the main circuit board that drives the USM motor.
You need to fully disassemble the lens and visually to judge where the problem is.
From what you described:
"To repair it's visible problems I glued the the cracked barrel and sand down to live the rings rotate freely. It worked and all mechanisms work OK because the bend did not affect enough to stuck the insides."
It looks to me the previous owner did not try to fix it. Or not try to replace the backward lens groups with some compatible version.
What you got should be all the original. Usually, the lens track for focus groups to be on track with the middle barrel; then the middle barrel on track with the outside zoom barrel should be unique and no ambiguous. You never know what the previous owner did with the lens to cause those out of track problem. I am very happy to hear by adjusting the mechanical track and you fixed the problem. Congrats.