According to the teardown, the Steam Controller is relatively easy to take apart with only a T6 screwdriver. You would need to reach [guide|52578|Step 5|stepid=116697|new_window=true] to get to the thumbstick (carefully unplug the ribbon cables on the trackpads before removing from the chassis). That should at least show if there are any physical obstructions causing the behavior you observed. Brush around the stick with a cotton swab dabbed with some isoprophyl alcohol.[br]
[br]
Worst case scenario, the thumbstick is soldered to the motherboard, so it would need to be desoldered to replace it. I'm personally not sure which would be a suitable replacement module for Steam Controller.