While the developers have not responded yet, I would like to add some common knowledge about the “state of the art” in this field. I hope this will make the discussion more constructive.
First of all, AFAIK, updates on the official F-Droid repository can’t be simply uploaded by the application’s maintainer. To ensure, that a published binary really corresponds to source code offered by application’s developer, the repository’s maintainer must download sources, build an application and sign it with his key. Obviously, all of this is periodically done by hand for every single application you can find on the official F-Droid repository. And, obviously, it introduces huge delays for update’s publication.
There is an experimental workaround involving reproducible builds, so that developers could build and sign their applications themselves and have their builds automatically verified by a build-bot before publication on the repository. But, the build-bot is still buggy and sometimes fails for unknown reason.
So, many developers created their own F-Droid repositories so that their users could receive updates as soon as possible. There are already dedicated repositories with Jami for various Linux distros, so, I don’t see any problem in creating another one for Android. But maybe I am wrong and I don’t know something.
Google Firebase and Push notifications
As it is described here, Google Firebase is used to wake up an application on new messages. Without it an application “sleeps” until user opens it, so it can’t react to anything trying to interact with it from outside world. IMO, this is a clear example of a vendor lock.
This issue is not unique to the Jami application. The Telegram-FOSS (a cleaned fork of the original Telegram application) also had this issue. So, they just show a notification all the time so that their application could run in background without being forcefully put to sleep by the Android OS. They also clearly explained this issue to their users. They even suggested how to cope with this distraction: just hide this notification. IMO, such publicity is the best approach to deal with this kind of artificially imposed issues, because reputation is still valuable for large corporations. (I prefer to naively hope that it is true.)
As I can see, the current F-Droid release of Jami (20200810-01) follows the same approach, but I have not found any description of this feature. So, the always appearing meaningless notification just looks strange. For comparison: users of Telegram-FOSS can open a detailed description of this workaround by just tapping on this notification.