First of all, that is an awesome workout.
Box squats train the hip flexors and when done on a speed day or with bands or both they build explosiveness that carries over to a more explosive back squat. Cleans build full-body explosion that carries over more to athletic and real world activities. Both lifts are good, but there is a difference.
Cleans are not a bodybuilding movement designed to target a specific muscle, you are supposed to use force and momentum. If you are not a competitive olympic-style lifter, the main reason for cleaning is to build explosive power that carries over into activities like exploding out of your stance in football. Due to the fact that the second pull of a clean is a fast, violent shrug done as high as possible, you will get a world of upper back/trap development.
As far as your elbow pain, I would have to see a video of your form on cleans to say whether or not that looks like the culprit, but if you think lowering the weight is hurting it, make sure to check your form. The correct way to lower weight to perform reps on the clean is to drop the weight to your waist, and bend at the knees to absorb the shock, then lower to the floor. If it is a 1 rep max and you aren't performing another rep, just let go when you're done and drop it from your shoulders, this works great with bumper plates, but if your gym doesn't have them, it makes A LOT of noise.