> 2) to provide objective proof for the true transaction history, anchored in energy expenditure.

Why do you need this if you are willing to trust other people not to steal coins or lie?

> 1) to fairly distribute all coins

Same question as above. If you don't care about perfidy, simply use the honor system for coin distribution.

If you do care about perfidy, then you should probably care about people breaking the law to steal your coins.