I wondered about that, but Bitcoin mining is increasingly dominated by a small number of big players, and they also have the hardware to do exhaustive searches

I wondered about that, but Bitcoin mining is increasingly dominated by a small number of big players, and they also have the hardware to do exhaustive searches on hash codes to find inputs that give the answer they want. I'm not saying its necessarily a bad idea, but its not immediately obvious that this is going to be harder to subvert than a selection of national lotteries.