Since you're "down under", I can understand not wanting to send the boards out. As to why the connector burned, this is a well documented problem with Indy - the engineers pushed the limits of what the pins could handle, and after time, the pins start to oxidize (corrode), which lowers the amount of conductive area on the pin connection, which in turn pushes more current through a smaller area, and higher current produces higher temperatures, which in turn leads to more oxidation, and the circle continues until they are at a point where the current flow through such a small surface area burns up the connector. Since this is not instantaneous, you may not have noticed a burning smell.
Looking at your picture, those header pins are burned - they will not have adequate surface area for expected current flow using incandescent bulbs. I know you have LEDs installed, however, you're taking a chance that there's enough conductive area on the pins to keep from burning up your connector. So, its a blanket recommendation that the header pins be replaced, too.
You may be able to take a scotchbrite pad or other mild abrasive and get the pins to be more conductive, but since the tin has burned off, they will become oxidized/corroded rather quickly. At a minimum, you will end up chasing your tail by doing the abrasive method lots of times, or at worst, the burning may get so bad as to burn a hole in the circuit board itself. In my experience, once you smell the magic smoke, the damage is already done.
Concerning the voltages - yes, you should be able to read the volts (I don't remember if it is AC or DC), but if you try to read the voltage before changing the header pins, you will find that you really have to have the meter lead "dig in" or scratch through the oxidation to feel the voltage. That is what a new connector would have to do to make the connection to a burned header pin.
I'm not sure of a source in your area, maybe others can chime in.