Earlier quoted context omitted.
It is impossible to create a "useful" PoW. The issue comes down to incentives. As bitcoin is a decentralized, trust-less system, there needs to be constant and powerful incentives to prevent miners from cheating (by creating forks, or executing any number of other attacks). The current main incentive against cheating is that miners will lose their invested PoW (compute time + electricity) if they cheat and get their…
> The current main incentive against cheating is that miners will lose their invested PoW (compute time + electricity) if they cheat and get their block thrown out. However, with a useful PoW, this calculus changes. Now, the PoW is no longer "wasted" when the miner cheats and gets caught as the PoW is now useful for something besides mining. So you spend $x on (compute time + electricity) to mine a useful PoW worth $…
To not care about the $z, $y must exceed $x by itself and that seems really unlikely for the types of things that OP suggested instead of proof-of-work.