The spinning hum, I'm sorry, but thats ridiculous. Thats not even a thing. Recordings are not picked up from inside the computer, they are picked up by microphones outside of the computer. And there is little to no noise interference with computer parts, they are all very well sheilded.
You can easily extract individual sounds from a recording, and a fan hum would be the utmost of easy to take out. Since tracks are illustrated as sine waves, a fan hum would be a constant straight line, occasionally going up or down. It would be one of the easiest tracks to identify and pull.
And if they can afford that case, they can afford a soundproof room.