Google authentication is listed as a feature of the Basic plan: Discourse pricing | Discourse - Civilized Discussion. It is possible that Discourse have already configured the discourse.group domain in a way that allows for it to work. (Maybe verification can be done on the top level domain in a way that also verifies subdomains.) It’s also possible that you’re the first person who has tried to set it up.
The reason for the delay in getting your questions answered is because the Discourse team members who could best answer them are away at a meeting this week. I’ll be sure to follow up on it with them when they get back if this doesn’t get sorted out this week.