So you spend $x on (compute time + electricity) to mine a useful PoW worth $y outside the blockchain and an additional mining reward of $z. If you cheat, you don't get to collect the $z reward. Although your original $x are not completely wasted, you still only have $y instead of the $y + $z you could have had.
How is that not incentive enough?
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.
Edit: silly question on my part. y and z should be trivially interchangeable. Your argument appears to rest on the idea that the monetizeable scientific worth of the work would dominate the worth of the crypto token itself, since there would be limited value in an already-mined result beyond standard blockchain speculation - is that the case?
scrathes head
That just makes $x grow to $previous_x + $y.
Eventually we may be able to buy cloud compute by inventing a new coin with a proof of work useful only to us, buying some Etherium to back it with (so that there's some initial value to mining it), and then letting speculators put it in there currency baskets to give it long-term value.
May not scale in 20 years. Useless research can become very valuable in a decade or two.
Won’t a market simply develop where people can buy / sell / exchange tokens?
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. This means that attacks are much cheaper and much more likely to happen.
One interesting part about this is that you can consider PoW usefulness on a sliding scale. The more useful the PoW, the more vulnerable your coin is to attack. Thus, you can probably get away with a "useful" PoW if it isn't actually very useful. This is one of the reasons why most "useful" PoW schemes focus on something which is pretty much useless (such as finding weird primes and whatnot).