So, sometimes when you buy a game on a phone or tablet and not a computer, it gives you the option to "remote download" the game. If Steam is running in the background, it will download the game you bought. As far as I know, it only works on paid games, but is there a way to remote download a free game?
No. To install the game remotely it needs to be listed here. If it's not, you need to start download on the computer you have Steam installed.
I've tried to simulate what "remote download" button does but for games that never were in your library it fails and returns different code.
Fun fact: Click
Play Game on store page, click
Yes, Steam is installed, cancel download window and from now you can download that game remotely. At least works for Fallout Shelter