What you have just described is some form of “signed telnet,” not actually SSH. Verifiable, but not private, unlike SSH. The closest thing to that in implementation is S/MIME or PGP signed email.
The medium of communication (Flash) is never “privy” to your messages in SSH. To achieve that, you will need another set of spells. SSH, when used with public key authentication, uses two sets of keys: one set for encryption, and another one for authentication.
Besides, there is one more problem: host verification. Batman needs to know that Flash will deliver the message to the right Wonder Woman, and not to an impostor. The second set of spells also helps perform this verification.