OPs problem is he needs his crypto keys at all times, because without them, he cannot sign up for a new service AND enroll them all. Now he's forced to carry his backup everywhere, making it just as prone to loss as his primary. Valid point.
Alternatively, if OP could carry a key ring of public keys for all his crypto keys, his backup crypto key can stay home, or in a safe deposit, or buried in a virgin forest of maple trees. While U2F may use ECDSA, or whatever other public key algorithm, it is not possible to carry just the public key and enroll it, is it?
The following is not 2FA, obviously: To illustrate the public key point more clearly, when I sign up for Github, I don't need all my private SSH keys on hand. I only need all my public keys to enroll them, and a single private key to use SSH. If at any time one of the crypto keys is stolen, I can remove the associated public key and use one of the backup crypto keys instead.