"Any assumption that the aircraft can move forward is predicated on the fact that the treadmill cannot match the engine's output. This is accurate, but like I said, outside of the scope of the problem."
If this is assumed then you are absolutely correct. This, however describes a situation that is different than the reality. If the problem is phrased "Assuming that the treadmill can provide opposite forces sufficient to prevent the airframe from achieving enough airspeed to fly, will the airplane take off?" Then no, the airplane won't take off.
Your example of the floats is a good one. The friction created by the water flowing over the floats would be much greater than that of wheels on a treadmill, so in that case the plane may or may not be able to achieve enough airspeed.
If this is assumed then you are absolutely correct. This, however describes a situation that is different than the reality. If the problem is phrased "Assuming that the treadmill can provide opposite forces sufficient to prevent the airframe from achieving enough airspeed to fly, will the airplane take off?" Then no, the airplane won't take off.
Your example of the floats is a good one. The friction created by the water flowing over the floats would be much greater than that of wheels on a treadmill, so in that case the plane may or may not be able to achieve enough airspeed.