Very good thread. Thanks to everyone who took the time to create well thought-out contributions. Before getting into my own recommendations, I think it’s important to review the background.
The Soviet Union was not expecting a German invasion. Therefore, the Soviet military was in a purely offensive posture. Also, Germany was more ready for war than Soviet military planners had realized. This created a window of opportunity, during which Germany was able to achieve a 10:1 exchange ratio against Soviet soldiers. (10 Soviets killed or captured for every German.) However, that situation didn’t last forever: the Soviets eventually got their act together, after which the exchange ratio declined to 3:1. At Stalingrad, the Soviets achieved nearly a 1:1 exchange ratio against the Germans. However, there were a number of times when General von Manstein achieved a much better than 3:1 ratio against the Soviets, well after Barbarossa had ended. For example, in the Third Battle of Kharkov (fought in the aftermath of Germany’s defeat at Stalingrad), von Manstein and the Germans achieved a 10:1 exchange ratio. The Soviet Union had a prewar population of 169 million, compared to just 69 million for prewar Germany. The Soviets could afford many more losses than could the Germans.
Any operation against the Soviet Union necessarily had two phases. During phase 1 (quick gains), Germany’s main objective should have been to advance as quickly as possible wherever possible, while capturing or destroying as much Soviet strength as possible. Phase 2 begins once the quick, easy gains end. Phase 2 would be slower and more deliberative than phase 1.
The gains Germany could make during phase 1 were limited not just by the Soviet military, but also by Germany’s supply situation. Germany was a coal-rich, oil-poor nation. Romanian oil production helped offset that, as did Germany’s synthetic oil facilities. But ideally from an oil conservation perspective, Germany’s soldiers would be supplied via coal-powered trains, and horses carrying food from train drop off sites to the soldiers in the field. However, the Soviet rail network was far more limited than Germany’s, the Soviets used a different rail gauge than the Germans, and Stalin had ordered Soviet rail lines destroyed as part of his scorched earth policy. The Wehrmacht was therefore far more dependent on petroleum than it would liked to have been. Germany had only enough petroleum for 2 - 3 months of active operations, after which its operational tempo would slow due to lack of oil. Lack of oil also implied an inability to deliver to German soldiers the things they needed: ammunition, food, medical supplies, and winter uniforms.
I agree with the OP that the Caucasus oilfields were of absolutely vital importance to both Germany and to the Soviet Union. However, Germany’s reach during Barbarossa was shorter than Germans would have liked. (Due to lack of oil.) The closer any given objective was to the front, the easier it would be for Germany to take. The Caucasus oilfields were considerably farther from Germany’s “starting line” than were any of the objectives it actually took during Barbarossa. The conquest of those oilfields would have fundamentally altered both the German and Soviet war efforts. But the capture of those oilfields was not an achievable goal for 1941.
Hitler had initially chosen to de-emphasize Moscow as an objective, preferring instead to focus on territory in the southern portion of the Soviet Union. That southern advance proved fruitful, and resulted in the capture of large numbers of Soviet soldiers. It did not, however, prove decisive. Hitler later changed his mind, and decided to go after Moscow. In an operation such as this, it is typically better to commit to one objective, than it is to vacillate between two different (individually tempting) options. Moscow almost certainly could have been taken, had taking it been a central focus from day one.
I’m skeptical of claims that the capture of Moscow would have resulted in the capture or death of Stalin. Yes, Stalin had remained in the city after the government had been evacuated. But I think the theory there was that it was easier to quietly evacuate one man than a whole government. My understanding is that there was a contingency plan to evacuate Stalin from Moscow if the situation had required it. Had Stalin been killed while trying to evacuate, that obviously would have been a tremendous bonus for the Germans. But the chance of that happening was probably well under 50% even if the city fell, so they would have been rather foolish to rely on it.
However, the capture of Moscow would have deprived the Soviet Union of a major source of industry and of population. It would also have been deprived of its most vital rail network hub; making it extremely difficult for the Soviets to concentrate their soldiers in preparation for an offensive. From a military and industrial perspective, Moscow was the single most valuable target Germany could have taken in 1941, even assuming Stalin escaped. There was also a good opportunity to have taken it in '41–an opportunity which would no longer exist in '42. Germany had also wasted an opportunity to take Leningrad in '41. Once that opportunity slipped away, Leningrad became much better-defended.
Germany needed to have come away from phase 1 owning Moscow, Leningrad, the Ukraine, and a lot of other territory in the western Soviet Union. With that territory in hand, it would then have been well positioned to launch a good summer offensive in 1942.