Linux

I mean, why not use a NUC pc? They will consume a bit more power but will run everything you need. I don't stream local 4k HDR content but I'd assume an Intel N150 chip or similar should be able to suit your needs. I've got one running for local cloud, websites and databases and it was just as cheap as an rpi5.

Does it have to be an SBC? I've had great success with a second hand Lenovo Thinkpad

I've been using a Vero V from the OSMC project. It runs Kodi and comes with a remote that's simple enough for my family to use. Kodi has been around for ages and is great for both local library content as well as add-on based content. The OS is based on Debian so you could run some other services on it. The SoC has support for AV1 hardware decoding (as well as all the older codecs) and can play 4K HDR10+ content to my TV without issues. There's also Dolby Vision support if your TV supports that. It's probably not the cheapest option but I wanted a small form factor and good UI that my family could use without having to be tech wizards.

Don't use a raspberry pi.

RPI5 only has h265 decoding, everithing else is handled by the cpu. Which is fine for 1080 as long as that's all the sbc is doing, but if you are also running some server, or you want anything h264 above 1080 you are out of luck.

RPI4 should be a little better, it has h264 and h265, don't know the supported resolutions/framerates, but the cpu is considerably less powerful. Also, the cpu lacks encryption acceleration, so if your are getting your movies over https that's gonna take a toll.

Older Pis are goint to be unsupported by kodi and jellyfin, so don't get those.

None of these is a dead no-go, listen to other peoples experiences. But I personally would advise against any Raspberry Pi. Maybe and Orange Pi is better? I don't know. My suggestion is to avoid the SBC, and get a cheap second hand Intel pc instead (possibly a very low power one). Intel's quicksynk video accelerator is gonna run laps around any sbc at any resolution, and it's gonna support more decoders, and even some encoders if you want to run transcoding in a jellyfin server.

Edit: If intel sold a quicksink pcie card, I would put one in my rpi5. But it don't.

Edit 2: I should add that some streaming services block 4k on Linux

Unfortunately with AI induced RAM pricing prices are up across the board so cheap is out if you want extra functionality.

You can get as someone suggested the OSMC device. As well as various other third party streamer devices that support OSMC or CoreElec though the Vero does hit near the top for features for video watching as per the spreadsheet linked here: https://forum.kodi.tv/showthread.php?tid=385600

But if you want general purpose distros that allow emulation you probably want an N150 or similar mini-PC. The tricky part there is that while you can get lets of cheaper older used mini-PCs for emulation if you want to do hardware 4K HDR video content decoding then you need something more recently made for efficiency sake so you're not pushing the processor to the max just watching videos.

If it has to be under $250 then your options are things like the Vero V or other streaming focused devices but you give up general purpose distro and emulation (I think, Kodi does technically offer a games feature but I've never used it and am not familiar) OR else you're going with a used mini-PC that will okay for general purpose distro and light emulation may struggle with your 4K HDR content depending on video codec used and bitrate.

If you can afford a bit more >$250 but <$350 you can get an N150 or similar intel system that should handle all of the above.

A caveat on the N150 route though is currently there is no support for Dolby Vision, HDR+, etc. You only get plain old HDR and that's not likely to change anytime soon as it's a driver issue and no one makes one.

If streaming your own video content is your most important goal and you may need things like DV and other HDR+ formats then go with the Vero V or another item listed near the top of the Kodi list like the Ugoos or Homatics.

My Rpi4 runs 4K HDR h265 content without problems, using kodi.

We have a now many years old Odroid that stills runs our media server, AdGuard Home, XMPP, and several other services.

It runs DietPi.

Your best bet is any recent-ish NUC.

Have a look at the QuickSync support list and check which hardware decoders you need. Then go buy a NUC of that CPU generation. i3 or i7 doesn't matter much, the decoders/encoders are part of the GPU and mostly identical within a generation.

Haven't tested it, but apparently the orange pi zero 3 is pretty decent at 4k.

https://www.youtube.com/shorts/cjVoLY4HN1U

Also got an LG TV. Why is their media player so shit? It would have been less effort for them to just fork VLC and then do absolutely nothing to it, never update it or look at it again. Yet it would still be so much better.

I know you said it must be Linux, but you could replace your Apple TV with an Nvidia Shield