Forget the TOS, it's certainly copyright infringement, since the licenses don't give you the rights to distribute game assets to third-parties.

A SAAS where people sign in with their own steam accounts (oauth), and stream games they own, would be fine though right

It's hard to say. Cablevision did win the lawsuit relative to their remote DVR service, which is promising.

I've been wanting to do this myself and prepared by registering several iterations of steam + host + box + cloud + remote.

The major annoyance is that since Steam doesn't let you do it outside your LAN, you have to use a VPN solution to make the remote server appear like it's local. I've been building my infrastructure and prepping for everything with the hope that one day they'll allow for the ability to enter a remote address.

If not, and when I'm ready, I'll just have to get my users to download some kind of utility that automates all the VPN functionality, graphics driver installation, and server-side management.