'sandbox site protected by login' ... does that mean there is an Apache login/password required before one can see a login screen?
I can speak only to Google ... not the others.
Google wants to talk https. One cannot setup a valid CA on a sandbox whose FQDN cannot be resolved. Check to see if you can install a plugin via Moodle.org. If you have issues with that, it's a clue cause even Moodle wants to talk https when installing a new plugin.
Will it show up when migrating to a live server? Well, hadn't tried that - so dunno, but, like I said, Google wants to talk https and it also has to be able to resolve your server FQDN.
There is a work-around for the certs that Moodle expects when installing a plugin. Moodle provides a cert for you to install in your moodledata directory.
See: https://docs.moodle.org/28/en/SSL_certificate_for_moodle.org
Section entitled: Provide the CA certificate manually
On the Google end assume you have web application credentials setup properly and have a key/secret that you've plugged into the config of Google on the Moodle server. One thing I did notice ... when copying from the lines that display the key and secret, one could acquire a leading space character for either/both and one could also have a trailing space character. Check to see if there is a space leading the key/secret and/or a trailing space for same. Obviously, remove the space characters.
Also check apache logs ... error log.
'spirit of sharing', Ken