Someone needs to go back to the pin manufacturer and verify their specs and production processes. When the pins mate, there should be a specified max contact resistance. There should also be some max variation in the resistance. With all the power wired in parallel, current should distribute itself automatically across all the pins. The problem is if the variation in contact resistance is too high, the low resistance pins will be asked to carry a lot more current than the higher resistance pins. This is where the overheating will occur.
It is important to understand it is the relative resistance of the pins that is important, not the absolute value. Two pins may have a spec contact resistance of <20 milliohms. But if one has a contact resistance of 20 mΩ and the other has a resistance of 10mΩ, the pin that has the lower resistance will be currying twice as much current as the other pin. And when we're talking about little pins and big currents, that can be a problem.