So I tried what you suggested. Well, I looked into the Big Picture controller options. Then I saw an option for other controller supports and turned that on.I wonder if that has anything to do with the controller configuration support that Steam recently added. Go into Big Picture Mode, then Settings and Controller Settings. Try turning on or off Xbox Configuration Support and/or Generic Gamepad Configuration Support. That could be interfering with the game recognizing your controller, or it could be that enabling it helps it recognize it.
I just remembered that I had trouble with the analog triggers on my MadCatz 360 pad in the original PC release of Psychonauts. Wouldn't recognize them at all, even though the shoulder buttons worked. I had to get very creative with button reassignment in order to play that game. I can't remember if later releases fixed that, or if I'd just moved on to another controller by that point.
Now it works PERFECTLY. So thank you.