tl;dr: Use SteamLink on a Raspberry Pi. It just don’t work on an Apple TV (4th gen).
I have a PS4 and I prefer play in the living room on the couch with a remote. I recently decided to finally use Steam a little more, because they have good deals on their store that are sometimes very cheaper than those on the Playstation Store. But I prefer play on the couch in front of the TV instead of near the Desktop PC, where I usually work, not play.
To do that, I can use the Steam Link service, to stream my desktop PC to the living room TV.
SteamLink app is available for both Apple TV and Raspberry Pi, and I have both.
Because I already set my Raspberry to play games, with RetroPie, I tried it first. I bought a Logitech F310 controller and plug it to the PI.
The app is easy to install on Raspbian, just search it in the “add or remove programs” section.
To play, you need to start the Steam app on your PC first. Then start the app on the Pi. I would like to be able to start the app using the controller, but I can’t.
So I need to use my Logitech K400+ wireless keyboard+touchpad to start the app.
Everything works fine, I can play for hours at Assassins Creed Unity.
But, because I also have an Apple TV 4th gen (model A1625, 2015), and I want make it easier to start the Steam Link for the family, I tried to do the same using the new Steam Link app for TVOS.
My network and host configuration is exactly the same for both client… wired network, a pc with an Intel i3 + GTX1060 gpu.
I can’t use any of the remote I have at home with the Apple TV, because, they only support MFI (Made For I(pod-pad-phone)) controllers. So, I bought the SteelSeries Nimbus from Amazon.
TVOS 13: you will be able to use ps4 and xbox controller on the Apple TV Soon…
The Nimbus is easy to sync from the bluetooth config, and automatically sync very quickly after when turning it on. Also, it’s easier to start the steam link app from the Apple TV menu, because I can do all from the Nimbus controller without using the Apple TV remote.
While connecting to my PC, the steamlink app show a warning about 4 buttons that are not available on the Nimbus: Select, Start, L3, R3. The default config for them is Menu+A,B,X,Y. I accept the default settings.
Then, I start to play. The same game, with all settings set to default, as I did with the Raspberry Pi.
Suddenly, the game starts to freeze. It locks for 0.5-1 second, every 10 seconds. And the sound start to cut. The resolution automatically lower, and it’s still the same.
I started the real-time monitoring of informations, and the fps were moving between 30 and 60, and suddenly switch to 1-2 when I get the freeze. And the status bar on the right show a huge peak.
I restarted all devices, and I went to the settings options, telling the steam link app to use “fastest” config, and I set the maximum resolution to 720p. Still same result, it’s unplayable.
You can set max resolution to 720p instead of 1080p. 
My conclusion is simple: the Apple TV 4th gen CAN’T be used with Steam Link. I don’t know about the 5th gen “4k” Apple TV.
The SteelSeries Nimbus is a good device, looks solid, and works very well with all native ios games. The issue is only when using the Steam Link app, nothing else. And it’s not related to the controller.
Because I bought the SteelSeries Nimbus ONLY to play Steam Link, and I can’t, I returned the controller to Amazon. But, again, it’s a good controller for all native TVOS games.
My suggestion is to buy a cheap Raspberry Pi 3 (or the new 4), and use any compatible controller you want, but I strongly suggest the Logitech F310, or a genuine Xbox controller (wired).