It is possible to design a proof-of-work scheme where being part of a pool doesn't provide any (or negligible) advantage, aside from the benefit of having a more predictable revenue stream. Perhaps that's what the OP is wondering about.
THAT IS THE ONLY ADVANTAGE POOLS HAVE NOW.
Each miner in the pool is working Independently. A pool with 1000 miners has absolutely no advantage over a pool with 10 or a solo miner (1). At its most basic level a pool is simply an agreement to "share" the reward. No work gets shared, only the reward. If a pool of 10,000 miners mines 5 million shares and then one miner finds the solution. Essentially the other 9,999 miners did ABSOLUTELY nothing. One miner solved the entire problem ALL BY HIMSELF.
However there is huge volatility in mining yourself (due to the high reward and low chance of "winning"). A pool normalizes the results (smaller reward, higher chance of winning) but the expected value is the same.
The protocols of the pool (i.e. checking low difficulty shares, rpc, etc) merely exist to keep people honest. The simplest pool requires no protocols just trust.
Me and 4 friends (all with exactly the same hardware) all hash independently. If one of us "wins" we split the prize 5 ways. Expected value is exactly the same but volatility has decreased. Now as pool gets larger the risk of cheating gets higher so pools implement protocols and procedures to ensure "fairness".
So back to the OP point. Please provide an example where the reward can't be shared by a pool.
You can't. Even if the work is completely non-parallizable the reward certainly can be via INDEPENDENT actors sharing the reward via mutual agreement.So what exactly would a non-parallelizable solution solve? Of course the fastest miner would always win the non-parallelizable solution.