Support

The email address(es) you get for Google Groups is for posting to Google Groups using a mail client instead of the web interface. Google Groups is essentially a hosted forum system. The emails you send to these addresses are not delivered in a mailbox anyone other than Google Groups itself can read. So, on that count, it's technically impossible to do what you say you want to do.

Reading your other tickets, you are trying to create a unified ticket system which collects tickets from several other sites, without creating a single helpdesk, and without creating a Single Sign-On. I will have to regrettably inform you that your use case is not something we consider feasible, let alone something we are interested in working on.

This sounds like a Rube Goldberg machine. You can achieve what you described here more easily:

Ideally, right now I'd at least like to receive an initial ticket from ATS via email and be able to treat that as the notification in Slack that also receives a copy of it, we still have to log into ATS to manage the ticket which is fine.

Set up ATS and the email fetch plugin as per the documentation. You can have a single mailbox in Google Suite which can receive email in several addresses (using a forwarder, or email account aliases). When ATS receives an email, it creates a ticket. When a ticket is created, all accounts with Support Staff privileges on the category get an email, as long as their Receive System Email option is set to Yes.

You can of course have a user account which belongs to a Joomla User Group that gives it the Support Staff privilege globally, AND a Joomla User Group which denies it the Frontend Login and Backend Login privileges. Effectively, you have an account which cannot be logged in with, but since it appears to be an active support admin gets emails for all tickets. Set that user's email address to an email forwarder you create in Google Suite. You can then add emails in the forwarded for everything that needs to be notified about tickets and ticket replies e.g. Slack. Therefore, you have a magic user which can be used for automating notifications. Isn't that what you actually want?