The second man (my uncle) would come out to the field on another tractor, grab the filled wagons, and haul them back to the crib. It was a nasty job: sitting on an open-platform tractor (no cab to keep the operator warm in cold weather) with very loud mechanical noise being fed into both ears from close proximity. We had a two-row corn picker, mounted on a tractor, that would strip the ears off the stalks and propel them into a wagon pulled behind the tractor. When I was a wee lad, we were in the last days of crib usage (at least on our farm). Those not interested in nostalgia should skip the rest of this.
The earlier ones are as you describe - wooden and rectangular. (At last, I get to use my Illinois farm roots in GQ. Field corn is stored in these on the cob with the shuck removed. Corn cribs are wire walled cylinders with a cone roof or a long shed structure with spaced slats on the sides for air circulation.