Why is SHA256/SHA512 not broken for shortish passwords I don't get why SHA-256/SHA-512 hasn't been broken, at least for shortish (10-12 character) passwords
It is perfectly plausible to pre-compute a password/hash-pair table using numerous very fast computers (thereby significantly increasing the speed at which this is done).
Clearly I am misunderstanding, so I would appreciate some input as to why this would not be considered breaking SHA256/SHA512 especially as most people use passwords shorter than 10-12 characters
SHA-256 and SHA-512 were designed to be very fast. Their primary goal ist to verify the integrity of long messages or files. Long means not 10-12 bytes but some megabytes and greater. It is not a good idea, to use hash function that is fast by design for password hashing. Instead, for password hashing should be used functions that need essentially more resources for brute-forcing, e.g. that require relatively much RAM, prevent or limit essentially parallelization and thus limit optimization of brute-forcing. For instance, Argon2, Lyra2, scrypt.
Compare it to cars. Ferrari is good for the cases when the speed is important. It is not correct to say that Ferrari is weak because it cannot carry big load. If you need to move a big load, you better use huge trucks. It is just a matter of choosing a proper tool for the task.