This would indicate a short in the circuit.
When I have seen repairs that began with these symptoms, it was a loose connection that connected two adjacent terminals right by the burned resistor in question. You would want to look at terminals 14, 15, and 16. If any adjacent pair has metal between them allowing them to touch, that would cause a short and the entire power supply voltage would be cooked of by the resistor you are referring to.
Since the other 270 Ohm resistor isn't exhibiting this behavior, we are able to narrow down the source of the issue quite specifically. (If it was a short in the upgrade, both resistors would be burning hot)