I think GOG could show Steam games. Probably used the same voodoo Lutris uses to display them.

Last time I used Lutris it would just download the installer from GOG's website which don't have any achievements unless they are accessible in the game. Heroic does have them but no notification while playing that you got one.

Steam in Lutris would just launch Steam so achievements worked there. But last I looked Lutris didn't display the Steam games you own through your family. And you had to actively refresh the list of games.