There are firms that sell phone scanning utilities that pull signal message off (unlocked?) ios and android (and presumably desktops).
Even if signal used an enclave key to encrypt the local chat database, sgx doesn’t protect the enclave from the keyboard, mouse or display drivers, so someone could simply write a screen scraper that displayed and captured each message of each thread.
I think you're misunderstanding what SGX is used for in the Signal context: It's only used server-side. The clients have no real use for it, because they don't have to attest the software they're running, only the keys they possess.
The server however is inherently untrusted, and the users of the server can benefit from some form of attestation of the software it's running. SGX tries to provide this, as the siblings in this thread explain.
Even if signal used an enclave key to encrypt the local chat database, sgx doesn’t protect the enclave from the keyboard, mouse or display drivers, so someone could simply write a screen scraper that displayed and captured each message of each thread.