I think you're spot on that many people don't even know what problems they've got. Something we'd like to do in the long term is build a community of solvers that actively seeks out projects. That way people are taking their expertise to the problem and saying, "can you use _these_ skills?".
Another idea is to have a team of enablers - if someone has a project but they don't know what to post we invite them to come and talk to the enablers in a chat room. They then help the project owner figure out what kinds of skills could help them, and get the project posted.
I loved that story of Toyota's engineers, and it was probably in the back of my mind when I conceived Solvers.