It's likely some mechanical problem with the innards of your bad switch, that is keeping the 'P' (pilot) terminal energized. The metal switch cover is just held on by the two tabs at either end: bending them outward a bit will enable its removal, so you can investigate. Needless to say, the switch will need to be de-energized and removed from the appliance first. Here's a webpage with many photos of an infinite heat switch inner mechanisms, which appear to be nearly identical to your own: https://www.electrical-forensics.com/ElectricRanges/ElectricRange.html

It's likely some mechanical problem with the innards of your bad switch, that is keeping the 'P' (pilot) terminal energized. The metal switch cover is just held on by the two tabs at either end: bending them outward a bit will enable its removal, so you can investigate. Needless to say, the switch will need to be de-energized and removed from the appliance first. Here's a webpage with many photos of an infinite heat switch inner mechanisms, which appear to be nearly identical to your own: https://www.electrical-forensics.com/ElectricRanges/ElectricRange.html

That said, the switch in the photo does appear to be a bit off of the OFF mark, and may still be in the depressed condition (which is necessary to turn on the burner). The knob socket could possibly be damaged, and not be indicating the correct position of the switch.