If the private key signature is introduced into the puzzle, in addition to the risk of losing the funds, the problem of cooperative mining still exists, this will bring greater probability fluctuations to miners.
The risk of losing money is mainly the leakage of secret keys in memory. There have been vairous ways of protecting in-memory sensitive data. You can find a brief discussion on them in the Discussion section of VRF-based mining paper.
In the case of combining secret keys into mining, the cooperative mining is no longer a profitable choice. Instead, the equilibrium is that each miner mines by himself, even though reward is unstable. Let me elaborate here.
When mining takes secret keys as input, the pool will lose his secret key if outsourcing mining. When a miner gets the secret key of the pool operator, he can steal money hold by this secret key anonymously by various ways (e.g., via mixing services). In short, outsourcing enables miners to steal money anonymously.
Then, consider a game between two miners A and B. A and B do not trust each other, but they want to stabilise their mining reward by mining together. When mining takes secret keys as input, the game goes as follows. If both of them collaborate honestly, both get most reward, i.e., stable income. If A collaborates honestly but B behaves maliciously, B gets all mining reward. Vice versa for A behaving maliciously while B collaborating honestly. If A and B behave maliciously, they are basically solo mining.
Thus, the equilibrium for this system is that, each miner mines by himself and gives no chance for other miners to steal his mining reward.
In this way, even though mining reward is not stable, cooperative mining is a worse strategy than solo mining.
Regarding your solution, joining mining pools is still the best strategy for miners.
There is an optimal strategy for each miner. A miner can try first phase for lots of times until getting a small difficulty. The first phase can be run concurrently. A mining pool can spawn more concurrent executions of the first phase, so have more advantage on mining than solo mining. In this way, pooled mining is still profitable, and miners are still willing to join mining pools.