Reading some of the earlier mailing list posts made by Satoshi back in 2008 and in his discussions with Hal Finney, it seems that his original vision of Bitcoin was quite different compared to the situation today. He did briefly mention ASICs as a possibility by referring to "specialized hardware" as well as the transition towards light wallets like Electrum and Multibit but he mostly referred to miners as being discrete individuals who mined using their personal computers. The only times where he talks about miners cooperating with each other to mine bitcoins is when botnets are discussed. Neither in his mailing list postings nor in his white paper does he ever mention the possibility of mining pools.
Each node's influence on the network is proportional to its CPU power. The only way to show the network how much CPU power you have is to actually use it.
If there's something else each person has a finite amount of that we could count for one-person-one-vote, I can't think of it. IP addresses... much easier to get lots of them than CPUs.
That quote is from a couple of months before he left the forum. Even then, he still held onto the belief that nodes are directly proportional to CPU power. Of course now we know that from the perspective of the network, someone that is mining on a pool is not considered a node and has no influence on the network. Instead, the influence lies in the operator of the mining pool.
Satoshi left this forum in late 2010. By then, the first pool (Slush's Pool) had opened and CPU mining was on it's way out. Perhaps Satoshi saw something that he had not anticipated and realizing what was coming, he panicked and decided to leave.