You can reset your password and redownload all of your messages to a new device if you use iCloud backup. Cook is full of shit when he says that Apple doesn't have the capability. They own the system.
Even a dedicated civilian could reset your password, associate a device with your account and receive all messages going forward. To state that Apple cannot is such a laughable claim that it becomes clear that it's just a PR game. Which calls into question how sincere he is in his feelings about privacy.
A civilian could still associate another key (device) to the account if they're dedicated with the password or a password reset (not as stealthy) assuming 2FA is disabled. And Apple could surely do it stealthily since they own the system.
They have the capability, and it's too kind to his statement by calling it a half-truth in that respect because it's really a lie when he says "[Apple doesn't] have the capability."