Let's say the winner of the block reward is decided by a lottery of some kind that forms part of the protocol. And in order to claim the reward he needs to do all the other - existing - things, which are currently external to the hashing process, yet considered part of the mining process.
Would that be a bad idea? If so, why?
Who runs the lottery? How is the decision made about who the winner is?
This is what mining solves. You don't need anyone to operate the lottery. The mining process IS the lottery. The process of hashing results in an unpredictable value. The miner repeats the process with a different input each time until he gets a winning ticket (a hash value that is lower than the the current target difficulty).
Additionally, by using the hashing work, it becomes impossible for an attacker to offer a different blockchain and claim that it is the real one.
Remember... When a node comes online after being offline for a while, that node needs to get the blocks it missed from its peers. If a node is offered two different blocks from two different peers, how does it know which is the "real" block? Mining allows the node to figure that out. It just chooses the chain of blocks that has the largest proof-of-work (which typically means the longest valid chain). To overcome this, an attacker needs to have access to more hashpower than the entire rest of the world combined.
Do you have an idea on how to run a better decentralized lottery?