If one looks at the board picture, it looks like the entire right side of it was meant to be a "break away" board, but never "broken off". That is because there is a grove cut all the way down the board, under those jumpers. All the jumpers are just that, to make connection with the right side of the board to the rest of the board.
That said, the burnt component does look more like a resistor than a white jumper. Though the best way to measure it is with one leg removed from the board, if indeed the boards were originally meant to be separate, it is likely the measurement across while in place "may" be valid. Lifting one leg to measure it is best, but one does run the risk that it might break.
Did the fuse shown in the picture ever blow? Was the fuse tested? That burnt resistor is close to the terminal strip that has the white/green stripe wire going to the fuse. Now if the fuse was blown, it is not likely the chair would move in either direction. Whatever cooked the resistor may have damaged other components, but I don't think this board is what actually drives the motor. There has to be some other motor interface I would think? Unless there is a lot of circuits under that grey cover like area.
In this picture, it clearly shows a resistor type of device in that burnt position:
Also, this image (upper right) seems to follow the labels on that board, the forward/reverse switch seems to be wired to that top right terminal strip, and possibly the burnt resistor:
Also, this manual seems to be for an updated version of that board, with schematics...