A proof-of-work or proof-of-stake system are the only known ways to stop someone from running a lot of client peers or hatchers in a decentralized system in order to block transactions or double-spend. I do not see any new solution to the Byzantine Generals problem in your answers.
proof-of-work = all the rich people control the damn currency!
proof-of-stake = all the rich people control the damn currency!
damn it, there has to be a way to guarantee an even distribution of control over transaction processing! But the only other way I can think of is if you could verify that every node involved in the verification of blocks is in fact a separate individual!