Talking about miners; the duties of a miner is not limited to just mining. Yea they get rewards for mining, but there are other benefits they gain by sticking around performing other duties.

Every time we initiate transactions, miners validate those transactions and ensure no double-spending occurs. They still go ahead to add this transactions to a block and then add the block to the blockchain. The transactions fees we are charged goes to the miners. Our only concern should be that when all the bitcoins are mined and miners no longer get rewards, they might likely shift their attention to transactions fees and charge higher fees. We really shouldn't bother about this because we might not witness this period when there will no longer be bitcoins to mine.

Maybe by the time these unmined bitcoins will eventually be mined, this guys can change their names from miners to something else that suits the job they perform in the Bitcoin network, especially in ensuring decentralization goes smoothly.