My suspicion is a shorted gas valve coil.
Brown (burned) wire provides 12V from control board (on back of heater) to both coils visible on the gas valve. This wire comes through a hole below, behind and to lower left of valve. Since this wire passes very close to both the upper (ECO/120V thermostat and 12V thermostat) and lower (120V element connections) plastic covers, if the brown wire got hot enough to burn, it would have scorched both covers. A shorted gas valve coil would have causes excess current and overheating of this wire.
If the problem had originated behind the upper cover, it should have burned completely through this plastic cover before the brown wire's insulation burned.
It was not a flame roll-out (flame backing up through the mixing tube). Ignition of the gas/air mix takes place at the far outlet of the mixing tube. Although hard to see on picture, your gas orifice and mixing tube inlet are below and well in front of the burned wire. Other wires and plastic-potted solenoid coils are between this inlet end of the mixing tube and burned wire, would have also been burned if roll-out occurred. These do not appear damaged. In addition, a flame roll-out would have caused flame-proving sensor at outlet (far end) of flame tube to lose rectification and gas valve would have VERY rapidly shut off gas flow.
There is no missing air shutter. LP-only burners are not fitted with a shutter (natural gas and convertible burners are, however). If gas pressure is correctly set (11 IWC), a venturi effect draws in exactly the correct amount of air for complete combustion.
"spiders" are a problem when they block the tiny orifice that meters gas into the mixing tube. This orifice is at the inlet end of the mixing tube. Blocking the orifice prevents the ignition of the air/gas mixture (either there is no gas available, or the mix is far too rich to burn). A flammable concentration does not develop because the flame-proving sensor at the tube outlet will closes the gas valve after a short "trial for ignition" interval. (It tries to ignite a total of 3 times only - the control module will lock the heater off (gas side only), and you have to cycle the 12V ON-OFF switch to clear this lockout).