Platform(s): Windows PC via Steam

Genre: Action/Shooter

Estimated year of release: 2000-2010

Graphics/art style: 3D third person shooter

Notable characters:

Notable gameplay mechanics:
I tried a demo of a game on Steam many moons ago, but I'm certain it no longer is on Steam.
Here is what I remember:

It was a third person shooter and early on in the demo you encountered this big rug hanging on a horizontal pole in a small backroom with many crates. You could shoot the rug, and it had physics calculations, that made it move believably.

[Fig. 1]

Then you encountered enemies in a huge room with ramps. Somehow you could take control of them or use your mind powers on them somehow. They had an aura to indicate this, which I think was a light blue.

[Fig. 2]

The same level also had a puzzle with ramps that were held up with hooks on a chain. You could mind-power those hooks to move them, so that the ramp would come falling down and let you walk up, or let enemies fall down.

[Fig. 3]

I know that many signs point to Psi-Ops: The Mindgate Conspiracy, and it looks close to what is in my mind, but I have not found any footage of the incredible rug, or the hook puzzle that I remember so clearly.

Update: It is not Mindjack (2011)