Talk about a winner-takes-all system, though!

You could spend a lot of money and time and do pretty well at approaching the problem, but still not make a living. Teams would surely disband and the members would all go into investment banking or fast food restaurant management, thus continuing the existing leaky pipeline problem. I think for big problems, potential prize-winning teams have to have some source (their own money? grants? bake sales?) of pilot-scale funding.

Look at the longitude prize. It's a contraexample to the notion that prizes are a great mechanism, since in the end the prize committee kept moving the goal lines and never gave the prize; on the other hand, John Harrison did solve the problem in the end, and he was supported with several grants of 500 pounds or more at a time during the 30 years it took him to do it.