It just doesn't exist. You will need some kind of mini PC like Raspberry Pi. I Use a mini PC from Beelink.
If you just want a media station with some basic apps a Raspi with LibreELEC is perfect. Your TV remote will work out of the box and there are plugins for Netflix, Disney, etc... available.
If you want a little more, like a working browser and / or some light gaming I would suggest a full OS with a little beefier PC like the ones available at Beelink.
How do you use your Beelink? More specifically what OS (and maybe core/most used apps) do you have installed? How do you interact with it (eg - wireless keyboard/mouse, USB IR receiver, etc.)?
Any downside to this approach compared to using the Smart TV/Android TV/Apple TV features?
My go to OS is Nobara KDE which is a gaming spin of Fedora. It has the best compatibility out of the box for my stuff. I'm sure other distros would word too tho.
Most used apps are Firefox for streaming, Steamlink for playing games from my gaming pc, Retroarch for psx and snes gaming and kodi to watch the content on my local network
Downside is obviously that it is less plug and play than a commercial media product. You need to set up everything initially, you need to update it and sometimes an update breaks something and you'll need to fix it
Upside is awesome tho: no walled garden. It's your gadget. You can do anything you like with it. No app is gonna track anything you don't want
I don't use jellyfin and I stopped using a raspi as media pc. My new media pc doesn't have an IR receiver so I use a mini bluetooth keyboard. While I used a raspi on my tv I had to dowload the correct IR package for the raspian distro I used and configured it. Or you could use Librelelec which has the IR support OOTB and check if there is a jellyfin plugin
I have a semi-decent Intel NUC that got decommissioned from being a game server, so it's not really in use.
Do you have any recommendations for a Linux distro or home media OS that I could set it up in the living room to play some emulators, watch Plex off my NAS, maybe some adblocked youtube or spotify, etc?
Pick up and play would be key. A wireless keyboard would be fine but a remote would be ideal
I don't know if the NUCs have IR receivers OOTB, which you will need to use a standard remote. I know the raspberries have them.
For best compatibility with NUCs I would just google for which distro to use. You're definitely not the first person to use them as media PC. I use Nobara on my Beelink and everything worked OOTB, but that's different hardware. Once you have a distro running you have all the options of a PC to run Plex, Firefox, Spotify, etc...