I always have to install the game and start it at least once. Then come back here, sync my games, and then it will be the last game at the bottom of the list or you can search for it by typing it in, but only after running it and syncing.

Yeah but I can't even see that list not to mention syncing games didn't even worked last month

I can, note that some games can't be on the list for some reasons so refresh and try again or clear cache

maybe you didn't set the steam profile public

Checked for my steam profile and it was public, already tried clean cache and I've tried to start a quest on another device like android tablet but all I can see is sync button and start quest but there is no list of my games nothing works

try login your steam from your brwoser first and then sync again, i have the same trouble and it worked for me