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 somet…
In a business environment, that's what you want. The key is then burned, and you ask your coworkers (who still have access) to remove the old key and store your new one on the servers.