Invite a Friend to Play a Game????

It kind of is. At least as far as the quest is concerned.
Anyway, the way it's been working for people so far is to pick a Remote Play Together game (you can filter those in your library using the advanced filtering options next to the search field) and invite someone through that.

Download "4 for the Money Demo" (~50mb) and invite to Remote Play Together.

Best game to complete this challenge is Brawlhalla. Me and my friends have completed this task with this game. Its only 419 mb in size and F2P.

https://store.steampowered.com/app/630/Alien_Swarm/ worked well for me !
Just make sure u invite any even offline friend to Remote Play . Steam will show that an invite was sent

Step 1: Open a game that has Remote Play Together (you can see if it has it on the store page)

Step 2: Open steam overlay

Step 3: Click on someones name

Step 4: Click on invite to remote play together

Step 5: Cancel invite (they don't need to accept the invite or be online)

Now you've completed the quest! :D

EDIT: You can search the game that has remote play via your library if you use a library search filter "Remote play together" ;)

Worked wonders, thanks a lot!

Serious Sam HD did the trick (Remote Play)