Make sure all your apps are up to date. Then:

sudo apt install steamlink

Then type steamlink

Then when you want to launch steamlink, press alt+ctrl+F1

Then type steamlink

Acknowledge the warnings for low video memory (doesn't matter)

Press enter the few times it prompts you

Then it will launch steam link

Connect your computer as you normally would

Then when you're done with steamlink, stop streaming using the menu in steam.

Then press alt+ctrl+F7 to get back to your OS.

I'm doing this on Raspbian on both my 3b and 4 pi.

The subsequent times you launch, you will only have to type alt+ctrl+F1 and then Steamlink, followed by 1 extra "enter". Use the F7 method above to get back out.

It took like 2 hours of random internet forum sleuthing to piece all this together. Hope it helps some of you!

Pro tip - Connect your Xbox controller to Raspbian and Steamlink will automatically recognize it. I have better luck with that than trying to connect using Steam settings.