As you guessed, there is no direct "voting".
I'm not exactly sure but I believe the mining reward is given out immediately. At least I think it could be given out immediately.
One of the main ideas about the consensus mechanism is that the network "decides" on which chain is the canonical chain. Basically it's always the chain which is the longest (or heaviest). As there is always latency in the network, different parts of the network have different perspectives and they don't necessarily have the latest information from the other side of the network. So different parts of the network may assume their chain is the longest - until they find out that there is a longer chain out there somewhere and they switch to that and abandon their chain.
So even if a miner gets his mining reward immediately he will just lose it if his chain gets abandoned (uncle). Yes, he could use the ETH as soon as he gets it but this is exactly the reason why exchanges require X amount of confirmations before accepting a transfer. A confirmation means a block on top of the block with the transaction. So if an exchange requires 10 confirmations before it accepts a transaction it means the chain with the transaction has to have 10 valid blocks mined after the transaction. After 10 blocks it is really really certain that the chain won't be reverted anymore (at least not 10 whole blocks).
So it's all about probabilities and the amount of security required. If someone accepts a transaction after 0 confirmations it's basically their own fault if the miner loses the ETH (in which case the ETH hasn't actually existed and the receiver has nothing).