More than that. It needs to be a problem that takes a predictable time to solve, it needs to be a problem that we're unlikely to discover faster algorithms for, and it needs to be something we can tie to a set of data.
Cryptographic hashes are designed to have these exact properties. If we wanted to replace this with a "useful" calculation, you'd essentially need to create a new hashing algorithm that also had some practical use beyond hashing a set of data.
Given that creating good cryptographic hashing algorithms is hard enough, I suspect that creating one that is tied to a "useful" result is not feasible.
Cryptographic hashes are designed to have these exact properties. If we wanted to replace this with a "useful" calculation, you'd essentially need to create a new hashing algorithm that also had some practical use beyond hashing a set of data.
Given that creating good cryptographic hashing algorithms is hard enough, I suspect that creating one that is tied to a "useful" result is not feasible.