I think you're confused about how the so-called "Bitcoin lottery" works. You seem to think that if I have some system and you have a parallel system with x100 the power, then you will find all the blocks and I will find none, because you'll always beat me to the punch. But no, these are independent Poisson processes (tied only via occasional difficulty adjustments) with different rates, meaning that you will simply find 100 times the blocks I will. So over a period where 1010 blocks were found between us, about 1000 will be yours and 10 will be mine.
In other words, it scales linearly - the amount you get out is exactly proportional to what you put in.
If that's all you're after, mission is already accomplished.
But if you think your "non-parallelizable PoW" system should behave differently, let's say that in this system a person with a computer finds one block per month. Then four people with a computer each should find a total of 4 blocks per month, right?. So a person with 4 computers also finds 4 blocks per month, because the system can't know who the computers belong to (and if it can then it's not at all about a different computational problem, but about using non-computational cues in distributing blocks). So a person with a special 4-CPU system also finds 4 blocks, as does a person with a quad-core CPU.
And, once more - pools
are not a security threat if implemented correctly. There's no reason the pooling mediator also has to generate the work. And, there are already peer-to-peer pools such as p2pool.
Edit: Parallelism means that an at-home miner can plug in his computer and contribute to security/receive rewards exactly in proportion to what he put in. Non-parallelism means his effect will depend in complicated ways on what others are doing and usually leave the poor person at a significant disadvantage (since others are using faster computers), which is the opposite of what you want.
In addition to the ones outlined in my above posts, I see one more: Currently the time for solving a PoW is distributed according to a Poisson distribution (Satoshi describes the consequences of this in his paper). We have a parameter (difficulty) where we can tune the mean of this distribution, but we cannot independently tune the variance of the distribution (with Poisson it will always be equal to the mean). With a different PoW system we will be able to obtain different distribution shapes (possibly with a smaller variance than Poisson). This could make the entire system more stable. Certainly it will impact the Bitcoin convergence behaviour. For the end user the impact might be a higher trust in a block with smaller waiting times.
Block finding follows a
Poisson process, which means that the time to find a block follows the
exponential distribution (where the variance is the square of the mean). The variance is high, but that's an inevitable consequence of the fair linearly scaling process.
If it pleases you, the variance of block finding times will probably be less in the transaction fees era.