If I understand correctly, this means you can't back up the private key, correct? It's in the Secure Enclave, so if you lose your laptop, you also lose the key? Since it looks like export only really exports the public key not the private one?

Probably not the worst thing, you most likely have another way to get into the remote machine, or an admin who can reset you, but still feels like a hole.

Or am I missing something?

ps. It amuses me that my Mac won't let me type Secure Enclave without automatically capitalizing it.

Edit: I understand good security is having multiple keys, I was simply asking if this one can be backed up. OP answered below and is updating their webpage accordingly.