Nodes are "vote by IP number", which is what Satoshi wanted to nullify by vote by PoW. Third page of his paper. There's no link between the number of IP numbers you control and your market stake. The last is not called "vote by full node" but proof of stake (which IS a sensible way to do things). Read the bloody Satoshi paper !
Who said anything about IP numbers? I dont even expose one; Im onion only.
What else does your "checking node" look like from the outside, else than a talking IP socket ? How do you hope that that talking IP socket is going to influence the Byzantine agreement ? How do you justify the claim that "a big majority of nodes imposes its will on miners/keeps miners in check" apart from counting them ? And what exactly do you count, apart from IP numbers ?
Who said anything about votes? Bitcoin is not a democracy. There is no vote by PoW; thats nonsense, and shows a total lack of understanding of what POW is and how it is used to achieve BFT transaction ordering. There is no vote at all.
If I present you two different block chains, which one are you going to accept ? The one with most PoW, right ? If that's not a vote, I don't know what it is. If you have a given group of miners, divided in two groups, each making their own prong of the block chain, which one is going to win ? That prong that accumulated the largest amount of hash rate (PoW), right ? It is because most PoW voted for that prong, that it won. That's the essential principle of Byzantine agreement with PoW: of all different proposals, the proposal where most PoW put its vote on, wins.
How, in case of consensus disagreement (two blocks propagated in parallel), consensus is reached again ? By the PoW vote of the next block. The following block is mined on top of one of both. So the block that got voted for by the next miner, and puts its weight of PoW on top of it, wins. The block that is not voted for, is orphaned, because its prong now has less PoW voting for it. If ever some other miners vote for the second block, there's no tie break, until one of the two prongs clearly wins the PoW accumulated vote. That chain that has most PoW voted for it, wins. And the property of PoW is that you can only use it once, that's why it is a "voting token": an amount of waste that cannot be re-used.
I guess that if even that is not clear, there's no point to discuss this further.