How do we know that a certain hash below the target is even possible when mining? If the target hash were artificially low, i.e. 3, how do miners know if there will ever be a nonce that produces such a hash?
For a given block header where the only variable is the nonce, it is trivial to brute force all nonce values and verify if any gives a hash that meets the target.
For the vast majority of block hashes, no nonce value will meet the target in normal difficulty settings (let alone at a target of 3).
Miners get around this by altering other values in the block header, such as the timestamp, block version, merkle root (via altering of the Coinbase transaction and/or set of included transactions).
By the balance of probability, you will eventually find one block header for some combination of the above that meets the target.
The difficulty levels control how likely it is for that block header to be found in ~10 minutes for a given hash rate.
The Bitcoin Wiki elaborates on how the calculation is actually performed to determine this difficulty, along with some example code.