There's not much that people in the GrapheneOS community can do to troubleshoot this issue because their app is proprietary and no helpful errors come up in logs that I can find.
But one thing I find interesting is this:
03-13 00:16:55.715 13395 13549 I FIAM.Headless: Successfully fetched 0 messages from backend
03-13 00:17:32.587 13764 14374 I FIAM.Headless: Successfully fetched 0 messages from backend
03-13 00:18:41.393 14166 14601 I FIAM.Headless: Successfully fetched 0 messages from backend
I actually had 15 of the same message. Always "successfully" fetching 0 messages even though there were messages waiting on the server.
I accidentally signed up for a trial work thing, so I am an admin for these two users and I can see history on the server. Some messages are sent my phone show up in the server's history but many are never uploaded. No messages were received by my phone from the server, including messages sent via the admin console.
Anyway, unless someone else tries and sees something in the logs that I missed, this particular issue would have to be looked at by the Zello app developers. They can test with their own debug builds and get much more useful information than we can from release builds.