I suspect you're missing one critical bit: miners get the transaction fees of whatever transactions they include in the block. They're mining blocks of transactions, not coins, approximately once every 10 minutes. Coins are just blocks that the mob of users claims are valuable.
This is true even after the last "coin" has been mined. Coins are incentive to jump-start the economy when there isn't much activity, the end-game is all in mining blocks to get transaction fees, at which point it pretty much self-balances. Miners continue mining as long as it's profitable, and transaction fees maintain a level which keeps it that way (but not too profitable, or there would be more miners competing for them).
This is true even after the last "coin" has been mined. Coins are incentive to jump-start the economy when there isn't much activity, the end-game is all in mining blocks to get transaction fees, at which point it pretty much self-balances. Miners continue mining as long as it's profitable, and transaction fees maintain a level which keeps it that way (but not too profitable, or there would be more miners competing for them).