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!
When the dozen or so startups who are working on a reputation economy find a way to uniquely identify everyone on the planet over the internet, it will enable a bitcoin competitor to make a decentralized initial distribution of the currency evenly to everyone, which would be VASTLY superior to bitcoin's system of mining as an initial distribution. Unfortunately, by the time this capability exists in one or two decades, bitcoin will have huge special interest groups who will attempt to destroy such a competing system before it gets started.
Mining sucks, but it is the best solution Satoshi had available.