Looking closer to the board connection of the 2 burned pins, they are going to one island only, so it seems that they try to mitigate the high amperage this way, using 2 pins. I hope I am right since I will be soldering the 2 wires together on that common island. Also the spacing between the soldered pins is rather small, so it will be a challenge.
And there's the rub. When one of the pin sets loses its conductivity (to cheap metal or poor tinning, or corrosion, or whatever) it stops carrying its share of the current, foisting it on to the other pin set, which can't handle it. This is a case of the engineering spec looking OK "on paper" and the cost looking OK to the bean counters, but then reality sets in, which doesn't care about either!
If you're sure two pins are the same connection, then don't try to join two fat wires onto the same circuit board trace. Join them off the board with a third wire of suitable gauge (I'd solder them), and then run that single wire to the board. That'll accomplish two things: the majority of the current will be flowing through the off-board connection, not through a circuit board trace, and you'll have a lot easier time of soldering to the board. If it's like the board I fixed, that third wire would carry next to nothing, only what the little components on the board might be using. The power-sucker, which is the SWG, will get its current through the off-board connection. If that's true of your board, that third wire can be much smaller, making the connection to the board that much easier. I have a drawing that illustrates that for my particular board. It's likely the same would apply to yours.
They do it the way they do it (using cheap connectors) in part to make the board more easily replaceable, but primarily to facilitate initial assembly and shave labor costs. You don't have that concern, so can afford the time to make a better connection. Again,
@generessler's suggestion to use connectors off the board is fine, but not more current-capable or corrosion-proof than soldering. Those bullet connectors are great, and time-proven in harsh conditions, but two of them utilize six
mechanical connections, soldering eliminates all six of 'em. It was mechanical connections that got you here.
If the following could apply to your board, you can make the repair quite easily, without removing/replacing the original connector and with minimal impact soldering onto the board. In my case, the big fat red and black wires power my SWG, the little black and red wires power the circuit board (so can be much smaller). The original green and white wires, which also carry little current, could be left as is. Now all the SWG's current flows through that splice, not the connector, and not the circuit board traces.
