Does this prove anything if it did, given that winds at altitude and ground level are vastly different?

The greater wind at higher elevation would only be an advantage to the hot air balloon.

It is impossible for any existing sailboat to have a downwind component that is faster than the wind. If it were possible, you could sail the boat with no wind at all.

No. If there is no wind at all, then if the boat moved forward in any direction, it would face an apparent direct headwind opposing its motion. But with some amount of wind, the boat can travel at some angle from dead down wind, so that the apparent wind will not be directly ahead, so that the perpendicular lift will have a forward component.

Will you at least agree that it is impossible to sail with the boat pointed directly downwind faster than the wind in a conventional sailboat (including racing sailboats)?

Yes.

A sailboat can reach faster than the wind because the mass of the wind is greater than the mass of the sailboat

Well, the total mass of all the air that was deflected by the sails as the boat accelerated past wind speed will be greater than the mass of the boat, but that does not really explain what is going on. The point is that the boat is able to continue to derive enough thrust from the apparent wind to overcome drag forces. And this continues to work even when a component of the boat's velocity is downwind.

And you cannot use apparent wind to explain moving downwind faster than the wind, because the apparent wind would be in the opposite direction.

It is not exactly the opposite direction, and even a small deviation can be significant.