So here is the idea:
Lets say we have a computational problem to solve and limited resources to solve it. The task would be to get as good as possible result for the problem within these limited resources. And every progress (getting better result from previous) would be gratified with "hash" coin based on proof of useful work.
Lets give a practical examples:
We want a video codec and best possible image quality on limited bandwidth, limited computational cycles on "virtual machine", limited code size and limited RAM resources. Library of wisely selected videos would be a reference for measurement of image quality. Every improvement in overall quality would be gratified with "hash" of the code. As long as final result fits within limited resources the algorithm of improving would not matter (machine learning/genetic algorithms ...) and be a subject of "manual" earnings.
Nearly same would be for audio codecs.
And then we would go for other important problems to solve for humanity - treatments for cancer, objects and voice recognition and so on...
What do you think about it? I will call it POUW (proof of useful work)
Currently if you were to buy enough hardware to attack bitcoin, for example, after the attack your hardware (which would have to be ASICs) are useless garbage, as the network would be devalued after attacking it and the hardware serves no other purpose. If I can then pivot to computing something of value, this economic hurdle isn't as expensive.
Also, the “POUW” examples you have are incredibly slow, developing codecs is going to take at least days, more likely weeks, months, or even years, meanwhile users will have transactions that need to be settled.
You mentioned CO2 in your statement about disliking cryptocurrencies but there are many currencies that do not utilize proof-of-work, instead using other schemes (e.g. proof-of-stake), I’m curious if you dislike those as well.
Specifically for the codec cases, it's not that we can't come up with better codecs, but that they require more computational resources to encode and even decode. We have ones now that are still a bit much for mainstream and using bandwidth/storage is a better market fit. As for the 'overall quality' this is subjective. Anything objectively defined is optimizing for something humans may not appreciate.
So what we're talking about is generalizing the hard to compute, easy to verify of finding hashes to other computations that are useful to some parties--a computational market. I do rather like this, like kickstarter for computation. A problem gets posted, with a verifier. People can choose which ones they want to sponsor and how much. Solvers can choose which problems to solve.
Even thinking this through, I think it's still useful to have a PoW hash-based blockchain just to record the first solutions. So we're back to being limited in the number of solutions we can record per second.
Because that's simply not how the world works. Not everything needs some grand purpose.
But let’s find some meaningful problems that come down to relatively simple calculations fist.