This assumption:

As no single party has the realistic power to find 2 solutions before others.

Is false. A 50% attack on the network is almost impossible, but a 5-10% is pretty reasonable. With 5-10% computing power you'll have quite a significant chance to find 2 solutions before others.

I see, thanks for pointing it out! It'd only be trustless for minor decisions, then, not worth risking the 1st hash reward to find another, and that's not very interesting.

This is related to how Bitcoin gambling sites can be made to be provably fair.

Is now private? I can't see it.