Full disclosure, I haven't yet made the switch away from the official client yet, but you could give Wholphin a try for android TVs. It lists customization as a selling point.
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I have a couple suggestions, the first may be obvious, but Jellyfin settings can be pretty convoluted so you never know.
What you may be looking for is the difference between “Up Next” and “Latest” (or “Recently Added?” can’t remember exactly, I already powered down for the night). You can add one or both as sections to your home page, which should keep all your latest additions front and center, while also keeping track of the stuff you’re currently watching, if you want.
My more advanced suggestion is to check out the SmartPlaylists plugin. It may not do exactly what you’re imagining, but it’s surprisingly powerful once you get used to the slightly weird UI and rules setup. You can build playlists based on very specific fine-tuned rules, so you could pretty easily set up playlists to only show what you want to see, and they automatically update as your library changes, which is pretty cool.
Accessing from the webpage, if you go to "Settings" > "Home" you can change the sections
This is what i have. "Recently Added Media" shows new additions for each of the libraries you have setup.

I also recommend Wholphin, it's faster and has better UI than the official Jellyfin app
To stop the updated items jumping to the front of the list I wrote this: https://gitlab.com/smaniottonicola/jellyfin-date-fixer.
It doesn't do everything you are asking for, but you may still find it useful.
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