Both run locally, so that is not the argument
CyberChef executes in the browser tab, which is exactly why it can be handed things that would never be acceptable to upload. This tool uses the Web Crypto API built into the same browser. Neither transmits your input, and anyone telling you otherwise about either is selling something.
What differs is the distance between opening the page and having the answer. A recipe-based tool is a superb environment for a problem you have to think about - decode this, then decompress it, then try it as a different character set. It is a long way round for the question of what the SHA-256 of a string is, which is the question most people arrive with.
Why MD5 is not on the list
MD5 is broken for any purpose that depends on two different inputs not producing the same digest. Collisions can be manufactured cheaply and deliberately, which rules it out for signatures, for integrity checks against a hostile party, and for anything security-adjacent. It survives mainly through habit and through file checksums published a decade ago.
Omitting it is an opinion rather than a technical limitation, and it is a defensible one only because the alternative is a click away: if you have a legitimate reason to compute an MD5 - verifying a download whose published checksum is MD5, matching a legacy database column - CyberChef will do it and this will not.
What hashing is and is not for
A hash is one-way. There is no operation that turns a digest back into its input, and a site claiming to reverse one is either looking the value up in a table of pre-computed common inputs or guessing. That is precisely why unsalted hashes of predictable values - passwords, email addresses, phone numbers - offer far less protection than people assume.
For passwords specifically, none of the SHA family is the right answer. Those are designed to be fast, and fast is the opposite of what a password hash needs. Use bcrypt, scrypt or Argon2, which are deliberately slow and salted. Use SHA-256 for content addressing, integrity checks, deduplication and signatures.