Mining is an all or nothing race. This is why miners join mining pools. Only a tiny fraction of the total hash power of the network is saved in the block that gets accepted. Maybe there's a more fair and efficient way to mine. What if miners could mine cooperatively, taking turns generating each block and helping to hash the same block.
Most miners join a mining pool to put the responsibility of maintaining the messy back end to someone else and just focus on their own hardware. That and if they are a smaller mining operation, they can choose to mine at a pool for a more regular payout. The mining profit is roughly the same before deducting the pool's fees.
When a miner joins the network the other miners respond by indicating their position in line and what position the new miner can take.
Mining happens in rounds. In each round, the next miner gets a turn to generate the next block- just the block data not the hash. They then allocate workloads to the other miners based on their speed. A workload is a range of nonce values. The miner who generates the block essentially hosts their own temporary mining pool.
As each miner completes a workload, they broadcast it to the network so that everyone can confirm it. The host credits them and supplies another.
Confirmation of work doesn't require re-checking every hash in a workload. The number of hashes checked at random can be based on the number of miners online. With every miner checking only a few random hashes, each workload gets checked a lot. If any miner finds errors in a workload, they inform the host who then scans it more thoroughly. Rewards are only allocated for actual work.
When a block is solved, it gets broadcast to the network as usual and the next round begins with the next miner in line assuming the role of host. If a miner drops from the network the round goes to the next miner.
The block reward and fees of the current block cannot be allocated among the miners in the same round so they are allocated in the block that follows it.
Unfortunately, Bitcoin network does not allocate mining this way. The block reward is claimed by whoever submits a valid block with sufficient Proof of work. There is no concept of individual miners on the Bitcoin network. You could probably do this queue system if you could allocate some central party to regulate this which could possibly idealize in an altcoin. Miners don't actually "check" hashes but it is a matter of bruteforcing and finding a block header hash that meets the current target.
