The main problem with this idea I believe is that SOMEONE must decide what problems to give to 'the work' being done. Bitcoin PoW works because it is just general work being done, and it is useful, not wasted, because it secures the blockchain, which is obviously of prime importance.
This, people see PoW as flawed because it "waste" computational power to solve the SHA256 puzzle but they miss the point. This energy is not wasted it is used to secure the network and makes it easily auditable. Eg: you know roughly how many watts of energy you need for a round of SHA256 which can be easily extrapolated to the network. This gives an ability to easily audit and verify nothing was tempered with.
There are something similar to what you described which is boinc distributed computer. You use your GPU for computation and get rewarded in "points" replace those points by token and you have something similar. However right now those point are worthless because they are just a way of ranking and you have no way to easily audit (by non mining node).
* How do you verify the work was done?
* In boinc it is centralized which means effectively you could print money