The stability pot will work either way because it's being used as a variable resistor. If it were a voltage divider (pin 3 connecting to 9volts) it could possibly cause a problem but most likely not. Worst case would be the stability pot wouldn't operate properly.
If I understand this correctly:
1) A fresh 9v battery supplies 9v disconnected from the pedal.
2) Connected to the pedal, it is 0.8v when pedal is on. Pedal seems to work, but LED doesn't.
3) Pedal off, you have 9v.
Your voltage is too low for the led to operate. Don't know why the pedal is working on such low voltage. But you mentioned, I believe, that it is weak after further review.
I would check the board, look at D1 and C5 for correct polarity. If they are correct, D1 and/or C5 may be bad. I would start here.
Let me know what you find.
Jim