The Germans didn't do much "wrong" aside than losing at Stalingrad, which forced them to withdraw the push into the Caucasus. With the threat of being cut off by the Soviet advance after Stalingrad, the Germans gave up the oil-producing regions around Grozny and Baku, and with a severely limited supply of oil for the rest of the war, they were doomed to defeat.
Even if the Germans took Moscow, it would not have harmed the Soviet war economy as the Soviets had already relocated most of their military industry to the Urals. The Germans were already stretched very thin and, without securing the oil in the Caucasus, it was not logistically practical to continue beyond Moscow.
Clausewitz was smart and his philosophies regarding realpolitick are worth paying attention to. But you can't cherry-pick one of his many writings and say that the capital should be the primary target for the invader without considering the context. Clausewitz was actually embedded with the Russian army during the French invasion of 1812, and he was there when the Russians burned Moscow to the ground for Napoleon to capture, leaving the French with a symbolic, but militarily useless victory.
Clausewitz was there when the French took the Russian capital, and he was there when the Russians refused to negotiate peace after losing Moscow, which was Napoleon's only option for a victory. Clausewitz was there when the French abandoned Moscow, starving and cold, and he was there when the Russians beat back the freezing, diseased, and shattered Grande Armée back to Western Europe. Clausewitz knows better than anyone else that capturing the capital city never guarantees success, if attempted under the wrong conditions.
In Clausewitz's own words (The Campaign of 1812 in Russia):
The Russians aren't learning from Clausewitz. In fact, it was Clausewitz himself who learned from the Russians, the value of trading territory for time.
Sorry for going off-topic. Just wanted to set the record straight.
The Germans certainly did much wrong beside losing at Stalingrad. For one thing, on strategic level they failed to maintain unity of purpose or consistency of objective in 1941. Their 1942 campaign leading up to Stalingrad again repeated the same mistake.