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.
That is not a correct statement. p2pool and eligius are evidence of that.
The pool creates a coinbase entry assigning the block reward to itself. When the block is solved, by whichever pool member submits a share of sufficiently high difficulty, that solution is transmitted along with the filled-in coinbase. Once accepted onto the blockchain, the mined BTC and fees are sent to the address that the pool assigned as the receiver in their coinbase entry.
if you are solo mining, you are creating a coinbase with your own wallet ID in it specifying where the reward goes (into your wallet) when you submit a block solution.