According to the segment I saw, typhus was the culprit, based on the DNA samples of the unmarred bones. And there were several mass graves between Russia and Europe that gave evidence of the disease.
We can survive cold, given the proper tools, but I suspect this was beyond what the retreating soldiers could handle.
In a way, he had the right idea — his entire host numbered something like 1.5 million, which is an enormous number. Obviously not all were soldiers, there was a remarkably well-thought-out support staff. About one-third of them made it back to the border of Russia. Horrendous. And some people still say Wellington beat Napoleon. :’)