@ogdento discovered that the two reds are connected to each other via traces on the circuit board. Same with the two blacks. So all we devised is to connected red-to-red and black-to-black
before they get to the board. Circuit-wise, the splices don't change anything. All the current draw of the IntelliChlor (IC) was running through the white connector and the traces in the board. While the traces are fine for that, the cheap pins in the white connector, and the some of the other parts downstream of the connector (the relay and the resistors, etc) are not up to the task.
As I said, my theory is the pins in the white connector don't work great. When new, they work well enough, but probably introduce a little resistance due to their poor physical connection and the cheap tinning they use. Enter the pool environment: acid and chlorine fumes, water, etc and the pins start to corrode. That corrosion leads to resistance, more corrosion, more resistance, etc. That resistance turns into heat. Soon enough the resistance builds up, the heat increases, and voila, melted white connector and fried pins. That severs the power connection to the IC, and it stops working.
I'm not clear why your other parts also melted. Though we have seen that before. Not all the IC current runs through those, but the excess heat generated in the white connector and the surrounding traces probably plays a role.
By cutting and splicing the wires away from the board, that creates a primary path for the IC's current that then doesn't flow through the connector. The connector still has some current running through it, but only enough to power the components on the circuit board, which is nothing compared to the current the IC uses.
We've yet to see this occur with an IC20. We know it happens with IC40s, and even Pentair warns not to use the IntellipH with an IC60.
Anywho, that's why the connector melts, and why the splice fixes/prevents it.