What is the point of allowing ssh-keygen to generate empty passphrase keys when it is not recommended to use such keys for remote login? What situations would such keys be useful for?
The passphrase protects the key in its storage on your local computer.
Even without a passphrase, a key is still better than a password, as it can only be "stolen" if someone has physical access to your computer (or at least some kind of network access to the files on your computer), whereas a password can be brute-forced (or leaked from any number of places in case you re-use passwords).