I'd like to point you to this excellent response by /u/killfile and build on it. I'd also challenge one of the premises of your question: the "key cities" were not "key" in the sense that they were indispensable to victory. Leningrad was besieged for a full year; it mostly functioned to tie down the Finns and Army Group North, and contributed little war materiel or Soviet manpower. Moscow, being the capital, was more important, but as /u/killfile points out, its function as a political center and railway nexus could be shipped east like so much of the Soviet Union's war industry was in '41 and '42. Finally, Stalingrad the city was not the objective - Maikop and Grozny were (i.e. the Caucasus oil fields, home to the majority of Soviet petroleum production). Hitler said, "If I do not get the oil of Maikop and Grozny, then I must end this war." Once Army Group South and Army Group Center reached the Caucasus and the Volga respectively (thereby cutting river traffic north and south of the city) the strategic objective was accomplished: the hideous street fighting to take the city itself was Hitler taking Stalin's bait and turning the city into a Kessel into which he poured men and machines.

The Wehrmacht objective in Operation Barbarossa was the "A-A line" (Archangelsk - Astrakhan). Subsequent operations may have pushed eastward to the Ural Mountains which figured heavily in Nazi planning, west of which would be the new Lebensraum, whose inhabitants would be starved out, and east of which would be the rump Soviet Union whose industrial centers would be bombed to smithereens by Luftwaffe air fleets, thereby reducing the Soviet Union's capacity for war and limiting the need for German troops and materiel to be stationed on the frontier. The Nazis expected the political structure of the Soviet Union to collapse with the onset of invasion. Hitler said, "We only have to kick in the door and the whole rotten structure will come crashing down." There was some discussion and a treaty with the Japanese of dividing the Soviet Union's territories east and west of the Urals between the two powers. I cannot quickly find a source of a unified German opinion on who thought what would happen -- there is plenty of debate even around whether Barbarossa's objectives were sound, but the general consensus is that the Nazis' assessment of the Soviet Union's political, economic, and social stability and war-making potential were unbelievably unrealistic. Both sides understood the war to be one of annihilation; as such, no serious coup attempts were undertaken against Stalin (and the NKVD was sufficiently effective that any such attempt would be brutally punished), because the cost of defeat in a war of annihilation is the ultimate cost--that of extermination of your country and your people; and that puts the cost of continuing to fight much lower than the cost of defeat.

So how close did the Germans come to victory? Beevor argues in Stalingrad that the outcome of the war was decided in December of '41 with the failure to take Moscow. If the Germans couldn't manage that, at the peak of their power, with the Soviets continually arming and sending divisions against them, the A-A objective was doomed; I'd go one step further and argue that even if Moscow was "taken" (taken? like, completely subdued? something that the Germans didn't manage even in Stalingrad city or in the Pripet Marshes or in a thousand other partisan-infested locales behind the line?), it would have been rather temporary in the face of the Siberian divisions being readied for action east of Moscow.

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Sources:

Antony Beevor, Stalingrad

Antony Beevor, The Second World War

William Shirer, The Rise and Fall of the Third Reich

Fuhrer Directive 21 available at wikisource

Omer Bartov, Hitler's Army

John Keegan, The Second World War