Unless the brown wire had insulation melted and the bare spot is touching something and when you move the wires to "Jumper" the Thermal cutoff it's either shorting or un-shorting I see not reason for the Fuse (for that's what it is) to cause the main fuse to blow.
IN "normal" operation it's a jumper wire. Just like the jumper wire you use to replace it.. Only time it is something else is if it gets too hot then it opens. Like a fuse. It's well Insulated by the plastic. Should not blow the fuse.
possible bad connection That can blow fuses by causing the device to "Stutter".