It can be done, but you'll always have a few ms of latency compared to a fraction of a millisecond if you do it over LAN at home. GPU instances runs on the latest Nvidia GPU's (NVIDIA Quadro RTX 6000) which should be able to handle most games.
You have to create a script that installs a GUI too, as it's not included by default. XFCE is usually easy to setup and it's lightweight. Steam is available in most distros repos.
Other than that, the only thing you'll have to consider is the price, because this is gonna be a very expensive setup starting around $1.5/h. It will be cheaper to buy or build your own gaming PC in the long term.