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I have a setup similar to what you want. I used ubuntu LTS minimal but you could use any distro that's has recent mesa and drivers. Kodi works well as a front end and you can use this to make your own add on to launch scripts from within kodi https://github.com/funcy2267/kodi-script-launcher
LibreElec only starts a Kodi shell and no display manager, you'd also manually have to get the dependencies for whatever you're trying to install because LibreELEC is very minimal. But I did find this Moonlight add-on that still looks supported https://github.com/veldenb/plugin.program.moonlight-qt. Steam Link on LibreELEC or similar is going to be unlikely, Valve only has it available as a flatpak.
One option I still have to try is Flex Launcher, in my current setup, if Kodi crashes for some reason, I'm stuck on a desktop that doesn't really work with my remote control or game controller and I don't keep a keyboard within easy reach.