If you think "that who gets the reward for a block, and who "solves" it are two different questions", then I am certain you do not comprehend that even in pool "mining" there must be an entity to receive the reward, what is not the pool participant. The reward needs to be transmitted (first transaction) to the entity which "solves" the block. You are failing to understand that the receiver (thus the "solver") it is not the pool participant.