Is the only use for the private key of a root certificate signing certificates issued by it? A root certificate can be used to issue intermediate certificates.

The private key of the root certificate is needed in order to sign any intermediate certificates.

However, after that, we only need the public key of the root certificate to verify the signature of intermediate certificates.

Similarly, the private key of an intermediate certificate is needed to sign the public keys of SSL certificates issued by it but after that we no longer seem to need the private key of the intermediate certificate for anything other than issuing and signing more SSL certificates.

Does the private key of a CA certificate have any uses other than signing the certificates issued by it?
It could be used to sign a Certificate Revocation List, for instance. Generally it is not used for anything other than certificate related actions, and some private keys belonging to root and intermediate certificates are therefore stored off-line to enhance security of the private key.