This is extremely inconvenient as it means if you're signing up for an account you need to have access to all those devices, even the one that's supposed to be

This is extremely inconvenient as it means if you're signing up for an account you need to have access to all those devices, even the one that's supposed to be in vault storage (potentially a bank vault, etc). TOTP just isn't a good protocol for this. Last-resort 2FA should be based on public key crypto where your last-resort device has a public/private key and you can enroll it as a recovery token by providing the (static) public key - this means you do not need active access to the device to enroll it.