Honestly probably not as Army Group South would be really stretched at that point. It’s objective was to secure the Caucasus with about half of its forces, while the other half took up defensive positions on the Volga and protected the other’s flank. While a drive onto Moscow probably would have been made by Army Group Center who was located near Rzhev. This was much closer to Moscow and the German High Command convinces Hitler to keep more troops there. The whole point of Case Blau or the attack on the Caucasus was to secure oil and the Volga river.

The Caucasus region controlled most of the Russian oil production, but Stalin would not have let the Germans get a drop of it. This oil would be desperately needed for any push onto Moscow. The other problem of Army Group South pushing north in a sweeping movement to Moscow is its flanks would be heavily exposed. Much of its forces would be worn down from fighting and it would need to supervise and protect the oil fields from constant counterattacks by the Russians. All Germany really had to do at that point is just sit and wait, as long as most of the oil fields were intact they would be fine.

It makes no sense for Army Group South to advance towards Moscow and expose itself as Army Group Center was primed and ready to do the same. But at this stage in the war, Germany only could have one Army Group advance unlike before where it could attack all down the line. This was due to its shortage of materials and resources, especially oil.