Second Steam Library as cold storage

It's probably a dumb idea, but hear me out:

I have only had M.2 SSDs installed in my computer for a long time now. But over time the games got bigger and bigger. When I'm in the mood for a different game or something big gets released, I have to juggle around to manage storage space.

Since Steam supports multiple libraries, I had the idea to simply put a second library on a network share and simply move all the games I'm not playing right now over there. And when I want to play it I simply drag it into my main lib. (pretty fast thanks to 10GbE)

This works really well and pretty much solved all my storage problems.

BUT there is one catch! I still can't drag a game over and immediately start playing after it's moved over. The problem are missing updates...

My idea: The main PC is not running all the time and also should not be responsible to update the fileservers games. But what about a tiny Linux container that has Steam installed and keeps the library up to date?

Would this be possible?
Can a library be shared between 2 Steam installs? Or is there a simple script to update it?

Would love to hear your thoughts!