They weren't able to win in the Caucasus because the whole place was a logistical nightmare. By the time 17th Army and 1st Panzer army were even close to the oilfields, they were already well ahead of their supply dumps, and as priority shifted to the fighting around Stalingrad, combined with the horrible terrain and growing soviet resistance, Army Group A was at the end of it's tether. Lengthy supply lines, which meandered through treacherous Russian roads in the Caucasus, were further aggravated by the difference between Soviet and German rail gauges, which made rail supply extremely difficult.

To make matters worse, a huge gap existed between Fourth Romanian Army, at the end of Army Group B's line, and 1st Panzer army in the Caucasus. Aside from the 36th Motorized Division, few troops were available to secure this part of the German line. Fortunately, the Soviets did not concentrate forces there either, they had operations going on further north (i.e. Stalingrad, Rzhev, Demyansk, Leningrad). However, come Operation Little Saturn, the Red Army was able to exploit the thin lines of Army Group A, and forced the Germans into a costly retreat from the Caucasus. In the middle of winter.

In many ways, Operation Blue marked the high-water mark of German offensives on the Eastern Front. The Germans went from launching a massive invasion (operation Barbarossa), to the assault on Moscow (involving three armies, three panzer groups), to consecutive invasions of the Donbas and then the Kuban/Caucasus region, to a 2-3 army attack at Kursk... which failed.

Of course, none of this should down play the incredible sacrifices made by the men and women of the Soviet Union, and the Red Army, in defeating the German invaders. I watched the episode as well, and I felt it did a fairly decent job of getting across the difficulties of mountain warfare.