A suggestion for HyperGG.com :
A button to launch the game depending on what game it is.
Let's use Rust as an example :

A button, in one of the corners, or close to the relevant game, that launches the game (Woruld only work for Steam support games) that launches and joins the server automatically. Simple. I think you'd need to find out the AppID then set the launch option to the correct IP address.
This may be what you're looking for if you think this is a good idea:
https://developer.valvesoftware.com/wiki/Steam_browser_protocolThanks!