Can they actually move their experience over to a new living room box thoug

I don't really see the need.

Valve is tied to the PC, which means that they don't have everything written to a single hardware platform, and their platform isn't locked down.

If you want a Steam console, get yourself an HTPC case, put whatever parts you want in it, throw some controllers on it, set Steam to autostart, and set Steam to use Big Picture Mode.

There is one company that makes Playstations. There is one company that makes XBoxes. There are zillions of companies that make Steam gaming systems.