Moscow was a bifg place in 1941 abd srategicaly vital, certainly to the Soviet Union. You make an interesting point about Japan though. If we have a situation where German troops have taken Moscow in October 1941 or at least are fighting in Moscow how does this influence thinking in Tokyo? While Japan is a member of the Axis war with the US was probably certain by this stage, Will Tokyo risk a two front war gambling they can defeat both the Soviet Union and the United States quickly. Another Japanese alternative might be to threaten a war and build up forces on the Siberian border, perhas initiate a border clash. However, last time they tried that, in 1939 at Khalkin GOL (Nomohan) Japan was badly beaten. Toky might wait until the Siberian divisions have redeployed to defend Moscow and hope to grab some territory as the Soviet Union falls (as Tokyo sees it) That however might be a miscalculation on their part which perhaps delays or even butterflies away Pearl Harbour - Japan, now at war with the Soviet Union pus off war with the USA and Britain until th Siberian campaign is over and Stalin has sought a deal. S we could have no Pacific War starting in 1941/42. The US is probably still dragged iinto war with Germ,any in this scenario, probably sometime in 1942 most likely over attacks on US escorted lend lease convoys. All this could have very interesting effects on the course of the war.
As an aside, I have always ignored the poster who you almost entered in a flame war for the very obvious reason that he seems like an old, stubborn man who cannot incorporate new ideas into his mind anymore. You don't respond to a person who responds to new, well-reasoned arguments by repeating almost exactly what he said before. It would be like arguing with a rock.
As for Japanese calculations, they had no interest at all in attacking Siberia. For one, Siberia had nothing worth conquering. While we know it is resource rich, extracting it with the technology back then was almost impossible. Japan's strategic focus was basically the oil and rubber of southeast asia without which its military would not be able to function. That is why it attacked the US over the oil embargo. Its military would be virtually ineffective within 6 months without a new supply of oil.
Now there is a strategic calculation to be had that the Japanese would attack Siberia in order to weaken the Soviet Union and allow it to fall to Germany thus helping all axis powers. But Axis coordination throughout the war was always terrible and every man for himself. See all the foolish adventures of Italy for absolute proof positive for this. This compared to the allies where America agreed with Britain that Germany was the larger strategic threat and to focus most of their resources on defeating Germany first even though their individual interest was in retaliating against Japan.
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Continuing back to the hypotheticals in this thread one where Moscow is taken easily and one where it drags down into urban warfare, I'd posit that it doesn't matter as long as Moscow is rendered ineffective as a transportation and economic hub.
The Soviet situation was quite desperate in the fall of that year. Tanks were literally rolling out of factories in Leningrad and Moscow straight to the front lines.
Lets consider why all the German Generals thought Moscow was so important. For one it was a transportation hub and in a world that was just becoming acclimated to cars and airplanes, railway was everything. It was especially crucial throughout the war because both the trans-siberian railway from Vladvistok and the northern supply railway from Archangel.
Without Moscow, supplying the factories east of the Urlas with American and British raw materials (especially aluminum which was in critical shortage since the main russian supply was from Ukraine) would've been far harder. In fact, the Finns knew they could cut off the Archangel railway link but never did as they didn't want to face harsh Soviet retaliation later if the war went bad.
And Moscow was still a large industrial center that accounted for about 15-20% of USSR industrial output. Much has been made of the Soviets moving their factories eastward but they were only able to do this for some factories and had to be supplied by America with railway cars, trucks, and other critical manufactured goods to make up for losses. Losing Moscow would've been not merely a symbolic blow but a significant logistical and economic one as well.
The fatal flaw in Hitler's thinking (other than the fact that he was ignorant on military strategy in general) was his preoccupation with economic resources. Any loss he inflicted on the Russians could be made up with British and American trade easily. For example, Operation Case Blue was premised on the fact that taking the fields at baku would cripple Soviet oil production. The USSR produced about 150 million barrels a year at the time. The US produced over a billion. Any losses, though significant, could be made up by the US in any contingency.
To defeat the USSR required crippling their transportation network as one of the prerequisites. The soviet counteroffensive with Siberian troops would've had to disembark hundres of km away instead of right at Moscow. American lend-lease aid would've been much harder to distribute. And logistically, without US trucks yet shipped in large numbers to the USSR, Russian supply would've been so crippled that the front likely would've been much farther back across the board.
Taking Moscow almost assuredly would've significantly extended the war vs the original timeline. IMHO, it's hard to see how Leningrad could've survived without the supply line from Moscow and if Leningrad falls, the Germans would have a port which massively improves their supply situation. Even without taking Leningrad, in the OTL they supplied hundreds of thousands of troops at the Demyansk Pocket which was a hooking line through the snow next to Moscow. The supply situation was indeed bad, but it's hard to say how it would've been worse had the Germans at least contested Moscow. Having access to some Moscow airfields and at least the ruins of housing is an improvement over the holes in the ground they dug outside of Moscow IOTL.
Over time maybe the Soviet Union still wins, but the Germans should've at least risked Stalingrad-like casualties in order to destroy the economic and transportation hub of the Soviet Union. They ended up taking horrendous casualties for some strategically worthless city on the southern Volga (at the real Stalingrad IOTL) anyways that had it been captured would've had no positive effect on the German strategic situation. If Moscow had been destroyed the same way Stalingrad was by intense street fighting, it would've been a humongous blow to Soviet strength.