jellycon VS jellyfin-apiclient-python

Assuming this is referencing the Kodi screensaver, then if I'm understanding the issue right, then this is one of those rare "feature not a bug" situations that's not a meme. There's a setting that exits playback when the screensaver activates. Depending on the age of your install, the default settings used to be that this was enabled. See this issue for more details. There may still be a bug here if it's not reporting playback properly, but I've never really checked that feature's behavior in depth myself.

In regards to the UI, if you're using a Kodi skin that supports customization (the default Estuary skin does not), then you can add your own widgets to the homepage. There's some basic guidance in this issue. While other skins may have different layouts and terminology, it's a fairly similar process.

JellyCon - "thin" Kodi addon, retrieves info from the server on demand

This is kinda what I'd assumed as well, but like I said i never tested. If you'd like, best way to remind me to look into it would be to open an issue on the repo: https://github.com/jellyfin/jellycon/issues.

There is an open issue for this, but I haven't had time to dive into it yet. https://github.com/jellyfin/jellycon/issues/181

JellyCon - Behaves more like a standard Kodi streaming addon. Media is accessed primarily by going through the Add-ons -> JellyCon menu, however you can set up menu options to link to it and show info on the home screen. It also allows easier switching between multiple Jellyfin servers or users since it doesn't have to rely on syncing all the metadata down. By not having metadata synced, it has to request info from the server which can take a bit more time when you're browsing (typically only a second or two in my testing), but you don't have to wait for the database to sync or keep it up to date.

This. Any skin that has customization options can be made to integrate with JellyCon just fine, and there's very basic steps for it in the opening comment of this issue.

2) You can try installing kodi and using https://github.com/jellyfin/jellycon, at least as a test. If playback is better, then the issue is the jellyfin androidTV app. If it's the same, the issue is (probably) the TV.

I'd recommend using the Python API client for Jellyfin, rather than trying to figure out the endpoints yourself: https://github.com/jellyfin/jellyfin-apiclient-python