> You want to give everyone a single vote so that the majority can agree on which transactions are valid and which are not.
That's not how Bitcoin works, although I admit it is plausible that you were describing how you would like Bitcoin to work. There's no "vote" in Bitcoin. Instead, there's a set of rules for determining whether or not a transaction is valid. Also, the distribution of hash rate does not necessarily have any relationship to the number of unique participants.
If you are building a system based on consensus you want to give everyone a single vote to vote for what is valid and what is not. But because Bitcoin doesn't want a central authority there is no easy way to ensure this by checking ID documents or something similar. Therefore everyone can create as many fake accounts as they want and let them vote for whatever they like. Bitcoin's solution to prevent everyone from casting as many votes as they like is mining, making it expensive to cast many votes. So the whole mining thing is nothing more than an elaborated fake account prevention mechanism without central authority.
I have some nitpicking to offer. I strongly disagree about the existence of any "fake accounts". All accounts are real accounts, even if they are being used in a Sybil attack. Arguably, the Bitcoin protocol should not care whether or not a block was generated by a "fake" account or not. Proof-of-Work as you acknowledge is critical for implementing distributed consensus in Bitcoin, but PoW has nothing to do with fake account prevention (and also, mining doesn't prevent non-miners from generating public or private keys anyway).
As for votes, I think calling it a vote is skating on ice too thin.... I ran into someone recently that also thought it was based on voting, but then it turned out that he was reading from something about PoS, which is not involved in Bitcoin's consensus mechanism.
I tried to just restate my previous comment but it seems I wasn't clear enough. Okay, once more. Account should be understood as user account, not bank account.
Forget about Bitcoin for the moment. We build a new distributed currency. You can sign up for an account and with that account you get one vote to vote for (blocks of) transactions you consider valid. But because there is no central authority performing identity checks the bad guys can just create a million accounts and use them to vote for bad (blocks of) transactions. This is what I previously called fake accounts - several accounts by one person to vote for bad (blocks of) transactions.
And now back to Bitcoin. Bitcoin's solution to this problem is mining. A vote in the Bitcoin network is a mined block and because mining is expensive this prevents users from casting arbitrarily many votes which is the logical equivalent of a fake account prevention mechanism. It is not perfect because with enough money for hardware and electricity you can still obtain more votes than others but it is no longer as easy as signing up for a million accounts with a million different email addresses.
That's not how Bitcoin works, although I admit it is plausible that you were describing how you would like Bitcoin to work. There's no "vote" in Bitcoin. Instead, there's a set of rules for determining whether or not a transaction is valid. Also, the distribution of hash rate does not necessarily have any relationship to the number of unique participants.