My Sega Master System might still be sitting on the shelf awaiting repair, but at least its controller can behave properly.
Button 2 was responding badly, so I opened the controller and cleaned the contacts and button parts.
I used methylated spirits in this 2013 video.
For electronic contact cleaning today I would generally reach for high-purity isopropyl alcohol or an appropriate electrical contact cleaner instead.
The bigger correction is that I would not now describe this as a method to apply blindly to every retro controller.
Controller designs vary.
Many use conductive rubber or carbon contact pads, which should be cleaned gently rather than attacked with an abrasive pencil eraser.
Inspect what is actually inside the controller first, clean the contamination and let everything dry properly before reassembly.
Thanks again to patrikw25 for pointing out the ENG/JPN switch behaviour in Wonder Boy!