Yes, unfortunately that is possible. Soldering by hand is never going to be as effective as the reflow method used in the original assembly, which would have eliminated any gap. It's also possible that a faulty component elsewhere, or a design error, is causing the FET to pass excessive current.Is it possible (or even probable) that if the drain tab isn't soldered on 100% right (I can't see how it wouldn't be, but there appears to be a micro gap about a fraction of a millimeter big between the bottom of the part and the pad on the PCB) that would cause it to go?
Perhaps. But less likely; especially if the original board had quite a long life before failing.Is it possible as well that the powerbrick they are supplying is too much for the board to handle when using NA designated electrical outlet?
It looks as though you have a fair bit of space around the FET. If it were me, I'd fashion a small finned heatsink from copper sheet, solder the FET drain pad to that, carefully bend up the gate and source pins if necessary, then solder the heatsink to the pcb pad and make the connections for the gate and source.