if the fact that the captain overloaded the aircraft and took off with a tailwind is true, it would seem that was one of the greatest factors in explaining why the aircraft crashed.
the aircraft was overloaded by something like 3 tons under the captain's authority with additional bags and fuel, and was taking off with a tailwind, presumably without recalculating the v speeds for that runway. the fire and the FE shutting down the engine was the icing on the cake.
other contributing factors included the fire, which could have been avoided if the airlines had followed recommendations by safety organizations to reinforce the tanks near the landing gear, as this was not the first time the concorde had an explosive gear failure incident leading to wing damage. in fact it had happened quite a few times throughout the 70's and 80's including several fairly severe wing damage incidents. the airlines elected to ignore the report and operate the aircraft as they were for almost a decade before the final accident. if the tank had not been full it probably would not have exploded as there would have been a small yet important gap between the fuel surface and the tank skin, allowing the empty space to absorb some of the shockwave that resulted from debris hitting the bottom of the wing.
a lot of things contributed to that accident. the dc-10 part lying on the runway was a big catalyst but certainly not the only thing that caused the end result. the whole takeoff was a mess of bad decision making.