The attacker doesn't need the key in order to decrypt ciphertexts or encrypt plaintexts since he/she already has the full implementation.
- So what's the point of hiding the key?
- What advantage would the attacker even have by knowing the key?
Update: ddddavidee's answer below, in the second paragraph, raises an interesting point: a white-box implementation of a symmetric cipher that exhibits the "one-wayness" property - i.e. it is infeasible to derive the decryption circuit from the encryption circuit, or vice versa - can be used to create an assymetric cipher. That would indeed be useful. Note that key-extraction security - i.e. it is infeasible to derive the key from the encryption (or decryption) circuit - is a necessary but not sufficient condition for achieving one-wayness.
However, I'm still stumped by the tenor of the first paragraph of ddddavidee's answer, which is why I asked this question to begin with. The model for an attacker of a white-box implementation assumes that they have full access to the encryption (or decryption) circuit. In that case (aside from the scenario in the previous paragraph), what is the point of the notion of key-extraction security, or key-extraction itself, for that matter, since the attacker can accomplish any encryption without using the key? Either I am missing something or there is something missing or flawed in the definition of white-box cryptography itself.