I haven't played yet, though I have downloaded. I assume this is the right place to flag a Steam issue. I've rejoined Steam after years of neglect; I have found EDCE aboard Steam but it's "not in my library." I don't see a way to add it to my library. Have you got a step procedural for this added layer? This is why I generally avoid Steam.
Viewing post in First Thoughts of EDCE.
At the same location where you downloaded the game, there should be a button that will provide you with a Steam Key. I think the key is like a 15 char long string (they use a "CD Key" format), which you need to put in Steam.
To do that, you open the Steam Client, and there is a top most menu bar, not the big font bar but above that. Click on Games->Activate A Product On Steam. Enter the key and you should be good.