I see the discussion going towards this particular question:
"What can I do in my training that will provide carry over to strongman?"
Correct?
I see Oly-lifting (cleans would be helpful in tire flipping, for example), strong compounds (bench, deads, squats) and conditioning work. Besides sport specificity type training (event work).
Basically, I look at strongman training this way: trying to improve my performance. ANYTHING that will improve my performance, I will train. Be it strength, speed, explosiveness.
BUT I don't believe you can do it ALL in one week. Which is why I train in weekly cycles; each week concentrates on a key area of performance. And I recently read that Juoko Ahola does the same thing. He sets one week aside for event training and one week for weights.