Mining is really simple: it's voting
It's really not. It's enforcing.
You can-not enforce without force. You choose to value transactions by the strength of the proof of work, but nothing is forcing you to do so other than a consensus.
Voting means there's a voting result that gets imposed on all the voters. In Bitcoin if two groups of miners enforce different rules both can operate, they just create a fork and thereby their own currency, that's all.
I'm quite convinced that mining = voting in Bitcoin is the biggest misnomer we have and should do everything we can to rot it out and not in the least try to perpetuate it.
I disagree with you quite strongly on this one. The thing is there are two separate, but related, issues here: transaction validity and transaction order.
Consider for instance the Iraqi dinar. I'm oversimplifying, but essentially after the fall of the Iraqi Government during the second gulf war there wasn't any force backing the dinar in the way fiat currencies usually are. However the dinar retained its value. Essentially a group of people had a consensus to follow a certain set of rules about the dinar, namely that pieces of paper that looked a certain way were accepted to have value. Even banking survived somewhat, thus in addition there was another set of rules that transactions following accepted accounting principles could be considered to be equivalent to those pieces of paper. This situation lasted until the coalition forces imposed their own will, essentially shifting this consensus to allow an additional rule: the coallition forces are allowed to make more dinars.
Of course all of this happened by consensus purely out of trust that the consensus would continue for however long you wanted to hold dinars.
Bitcoin is of course also a consensus, specifically that transactions that follow rules that the Satoshi client audits should be considered valuable. The thing is it is
not miners who enforce those rules. The consensus enforces those rules, simply because the consensus is that any transaction that doesn't follow those rules is considered worthless. It'd be theoretically possible, albeit totally impractical, for someone to decide to accept a transaction after carefully checking the whole blockchain by hand to determine if the rules were being followed. Bitcoin clients simply automate this process.
If you could somehow trust people not to double-spend you
would not need mining. However we can't trust people, just like we can't (blindly) trust people not to write checks that will bounce. Thus Bitcoin has an additional rule where we consider a transaction invalid if a transaction spending the same coins exists
and has a stronger proof of work. If we could somehow timestamp transactions directly, and in addition have some mechanism to be absolutely sure that everyone has an up-to-date knowledge of every transaction made, again we wouldn't need mining.(1)
Miners of course choose what transactions to include in their blocks, and that part of mining I don't (strongly) consider to be voting. Where the voting comes into play is which block they decide to use as the previous block for their block creation attempts. Sure the outcome of any individual block is discretized in a way that no normal voting system would be - only one lucky miner will solve the proof of work - but after repeated blocks the outcome is effectively a vote proportioned by hashing power. This is particularly evident with cases like non-standard transactions and new types of transactions, and we speak of a "majority" and a "super-majority" of miners for a very good reason.
There are strong parallels in how governments work: you have the law, which enforces how politics and voting is conducting, and then you have the votes themselves, which decides who gets into power. It's just that in Bitcoin takes the "first-past-the-post" effect to the extreme, so much so that changing the rules is exceptionally difficult.
Anyway, more to the point, we're talking about how to explain Bitcoin to non-technical people. Analogies are never perfect, and yes, the analogy that mining is voting certainly has flaws. However I strongly think that using the analogy of voting to explain what mining is, and laws to explain what the transaction rules are, makes far more sense to people
and gives them a much better mental model of how Bitcoin works than starting with a much more technical explanation that immediately jumps into the proof-of-work and enforcement. After all, saying that solving a difficult math problem somehow enforces something just doesn't make sense to most people. Saying that you solve that math problem to prove that your vote should be counted does. Your explanation is more technically correct, but the vast majority of people have a good intuition for what voting is, and no intuition at all for what a proof of work is.
1) As a thought experiment, consider what would happen if every Bitcoin client had a magical radio that could broadcast a signal so powerful that every other Bitcoin radio was guaranteed to receive it, and at the same time, everyone had access to one or more radio receivers in secret locations that they used to ensure they weren't just receiving a faked, directional, radio transmission. If everyone could leave their client running 24x7, and everyone had access to a secure time source, you could replace mining with simply the assurance that you would always know what transaction was valid simply because you were guaranteed to see the valid one first. Equally you would know you could spend your coins because everyone else could do the same verification as well. (bootstrapping your client does of course pose problems with this scheme)
