I definitely see your point and your ideas make a lot of sense, but both times I played, at least, the Italians were able to do relatively little against the Russians, as Northern Italy is 4 moves away from Bessarabia and in order to get a substantial Italian force on the Eastern Front, you need to divert valuable troops from the Mediterranean theater. When I played as the Italians, the only time I attacked Russia was when I had made a Minor Factory in Egypt and had a large army moving through the Middle East, which Russia tried to counter but failed in doing so. The only reason I was even fighting Russia in the first place was because I had pushed all of the British units out of the Mediterranean and Middle East, and the only south of England were a few infantry near South Africa. Russia had tried to make a last ditch effort to stop my advance, but ultimately failed, and if we had had time to continue playing the Italians would have most likely have taken Stalingrad from the South, as the Germans pushed up and around.
As Germany, my plan to counter the screen was to gather a large army in the South, in conjunction with the Italians pushing through the Middle East, in an attempt to take Stalingrad, which only had a minor and was therefore hard for Russia to hold onto against the advancing forces. The Russian player had a large stack of infantry in Karelia, and so I began to build a fleet of transports to have an amphibious assault with my units produced in Germany in conjunction with my Eastern Front forces (but unfortunately the Russian and English players decided to do a one-two punch air strike which sacrificed their planes but managed to destroy my 7 transports and other ships). I made a feint assault on the Baltic states, which he responded to with a counterattack by about half the forces he had in Karelia. While I took heavy losses, I was able to take it back with relatively minor casualties through landing newly produced units from Germany and my still powerful air force.
With the whole maneuverability aspect, as the Russian player I made sure to build large amounts of Tanks, Planes, and Mechs so that the situation you described can be avoided. The large amount of infantry I started with, I placed evenly behind the three screening territories, and I situated the Mechanized and Air Forces in positions where they could be sent to any front that needed them, and made sure I had enough blocking infantry so that I would have enough time to deal with one threat at a time.