Sure thing. I have actually been going through the documentation extensively as I was butting my head against this and I think that I may have finally found a solution (although, I am keeping an eye on the logs to make sure).
The main issue that I had been having appeared to be with Organization Audit Logs (organizations/${OrgID}/logs/
cloudaudit.googleapis.com%2Factivity) and the Log Sink that was created to route the logs matching the filter to my Pub Sub topic. From there the Subscription (Push) would be triggering a Cloud Function to get the gropupUniqueId from the protoPayload of the logs and make a Patch request to Group Settings API.
I hadn't been seeing any errors linked to permissions or any activity and I making sure that I was following this:
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But as I was looking, I also saw this bit, which is where I think I was encountering the problem:
When I got up this morning to do some more testing, I finally started seeing activity from the Writer Identity account that was created with the Log Sink so I think that this is what I had been running into.