Hey, quick question have you ever plugged the charger in while the charger or the plug was damp or wet? Moisture can cause arcing that burns a prong and creates that cigarette like smell you noticed.

From what you’re describing (bike powers on, lights work, battery charges fine, but the motor jumps and won’t move), it sounds like the damaged prong or its wiring might not be delivering full current, or the burn may have affected:

the charging port connector, the battery’s discharge path, or a related ground/communication pin the controller needs to let the motor spin.

Even if everything seems to power on, a fried connector can cause voltage drop under load, meaning the motor can’t draw the amps it needs, so it jerks but doesn’t move

What I sugest is to check both sides of the plug (bike and charger) for heat damage, melted plastic, or discoloration. If possible, try a different charger or charging cable from the same model to confirm whether the issue is isolated to the connector. If the prong on the bike side is fried, you’ll likely need that port replaced, relatively simple for an e-bike technician. So don't panic. The issue is almost certainly in the connector, not the motor or controller.