Hmm that is super weird... my bet is:
1. the grey coil is fried. i don't know why this happens but it was taught to me early on in mopeds and i have seen it being the case. it's not putting out enough to power the bulb, but the test LED has much less draw. if you swap the brake light power to the tail light assembly, does the bulb light up?
2. less likely, but there could be a smudged ground somewhere in the assembly where the bulb goes in. Doesn't fully explain the test lead tho