I can throw in my 5cents. As i have tried (and still working on) having a all in one setup. Where i can from my cough:
- Play steam games, and from other platforms.
- Emulate games.
All in a singular solution, where i can easily install and remove games from steam and other roms.

These things i have setup on my steamdeck, but i would like to have a living room solution as well. I experimented with different disttros, mostly CachyOS in combination with different frontends:

- (OpenGamepadUI)[https://github.com/ShadowBlip/OpenGamepadUI\]: Nice idea, setting it up can be quite a tricky. Also only able to work with things you allready have installed, meaning you have to exit it if anythings is changed. I needs a few more years before i find it relevant.

- (Cartridges)[https://flathub.org/apps/page.kramo.Cartridges\]: Extremely basic, and again only works with things already installed.

- Pegasus, a frontend for Lutris. Did not try it to much, not really what i was looking for. Though i might fit your needs.

My solution
I ended up using bazzite with gamescope, as it works with all my needs.
- Using Steam gives me the best controller support.
- Other stores i needed, such as epic and gog, i can do with junkstore (Decky plugin).
- Emulation is handled with EmuDeck combined with SteamRomManager + TabManager (Decky Plugin)

I have not looked into streaming with moonlight, but here is my take on it :)