Why can't miners meet the difficulty by picking a low number for the block hash? In terms of 'proof of work' my novice understanding is that miners are tying to come up with a 64-digit hexadecimal number, "hash," that is less than or equal to the target hash.

Can't miners always get lower than the target has by inputting "63 zeros followed by a 1?

Beginner here, I apologize for my simplistic understanding.
The hash is a result of computing two rounds of SHA256 on the block header. If you set the hash yourself, when other nodes try to verify the hash by performing the aforementioned computation, it will not (however, there is an infinitesimal chance) result in "63 zeros followed by a 1". I suggest reading about hash functions. 

The only reliable way to create a valid hash is to continuously compute the hash on the actual block header, while modifying the contents of the block so that the block is still valid (e.g. nonce, transactions, block version, etc.), but gives you a lot of tries (guesses) until you get an output that is valid. It is basically brute forcing, so having fast, efficient mining hardware is an advantage.

See also: Block Hashing Algorithm