The concept of “Proof of Useful Work” is flawed because if the work is useful, then it has a value and can be sold, and the value of the work reduces the cost of mining so that miners will increase their capacity. In the end the same amount of useless work is generated because that is the actual limiting factor.
For example, suppose:
1 POUW coin is worth $1. It takes 1 kW to mine 1 POUW per day and 1 kW costs $1 per day. However, the work needed to produce 1 POUW also generates $0.50 of useful work, so mining 1 POUW worth $1 has a net cost of $0.50. Obviously, the miners will increase their capacity in order to increase their profits. Suppose they double it. Now they consume 2 kW to mine 1 POUW worth $1 and generate $1 worth of useful work. Even with the useful work, they are still wasting $1 of energy per day.