Why is bcrypt the right choice for a password when SHA-256 is stronger?
Because they are built for opposite goals. SHA-256 is designed to be fast, which is what you want for integrity and exactly what you do not want here — a GPU tries billions of SHA-256 guesses a second. bcrypt, scrypt and Argon2 are deliberately slow and tunable, with a work factor you raise as hardware improves, and they salt every password so two identical passwords do not produce the same hash and a rainbow table is worthless.
The number that makes it land: the same GPU that does billions of SHA-256 hashes a second does tens of thousands of bcrypt hashes. That ratio is the entire security margin.