Exactly. Everyone knows you have to starve Italy, but not everyone seems to realize that you need to do it quickly. If you starve Italy quickly, then Italy has to choose between building a significant can opening force…they can’t afford to do both.
If Italy builds a significant can opening force, you can and should take Rome. You also take Northern Italy if they leave it weak, but that’s optional and not worth diverting an entire extra fleet. If you want a European factory, you can get it by taking Normandy and/or Southern France; you don’t need to base your entire strategy around protecting a factory in Southern Italy. If Germany wants to repeatedly send mechs south to Rome, that’s fine – you can set up a shuck-shuck from USA - Gibraltar - Rome, build a naval base in Sicily, retake Italy’s capital each turn at a profit, and soon enough your transports will start to complete successful round trips, saving you even more cash that can be spent in the Pacific. Transports go to Gibraltar turn 3, Rome turn 4, Gibraltar turn 5, and back to USA on turn 6, so starting turn 6 you don’t need any more Atlantic transports and you can go 80% Kill Japan with America. Meanwhile all those German mechs that died in Rome are no threat to Egypt. Moscow will fall hard, but you’ll have significant pressure on the western front and Egypt only has to fight on one front, so the Allies can still win.
If instead Italy builds mostly infantry to sit and defend Rome, then you can and should move the surface fleet and airforce to take Normandy and Norway while leaving your subs behind to convoy Italy. Italy won’t have the offensive punch to kick you out of France or help threaten Egypt in the endgame. Italy might eventually build a destroyer and whittle down your submarines, but that’s OK. If Italy comes back to life earning 12 IPCs on turn 8 after having bought almost all infantry and destroyers for the whole game, they’re still not going to pose a serious threat to Egypt before the game ends – especially because with no can-openers to weaken Russia, not too many German tanks are likely to be punching through the Middle East. You might even hold a British Persian factory the whole game if you use a Middle Earth type strategy. If Germany also builds mostly infantry to defend the western front, then Europe is safe and you go 100% for Japan to try to win on that side. If Germany builds an extra couple of rounds of mechs and tanks to try to push through to Egypt or India, well, you’re already sitting in Norway and Normandy, so you keep pushing there, take Denmark and Western Germany, and try to get Berlin. I think the Allies have pretty good odds either way.
That’s why I typically use a third strategy as Italy: I build fighters, subs, and destroyers to try to sink whatever boats the Allies put in the Med, I let Italy take Southern France, Yugoslavia, Bulgaria, and Greece for the extra cash, and I will use German planes to help clear the Med if I can do that at a profit. I can’t force the Allies to give up control of the Med – they can take it if they want it – but I can often force them to pay such a high price for it that it interferes with their ability to run a campaign anywhere else. If Japan is uncontained, Germany holds Normandy and Norway while still making steady progress toward Moscow, and Italy is alive on turn 5 despite finally losing control of the Med, then I think the Axis have a good chance to win. Even if the loss of German planes means you will lose so many tanks taking Moscow that you will never take Egypt, you can still win an economic victory – if you’ve got Norway, Normandy, Leningrad, Moscow, Stalingrad, China, southeast Asia, and the money islands, that should be plenty to win the endgame even if India is still hanging on by a thread and Rome is reduced to building infantry and waiting for the end. Germany is rich enough to hold off the combined US/UK in northern Italy indefinitely while still picking up a few territories here and there (Urals, Kazakh, etc.), and Japan is also rich enough to hold off all remaining Allies while continuing to slowly expand (Western Australia, South Africa, Siberia, etc.)
As a side note, I’m not a big fan of taking Greece unless you’re playing Balanced Mod; it’s just not worth enough money without an extra NO, and it’s not strategically vital if Germany is doing Barbarossa while Italy is camping at home. Maybe ABH can explain why he likes to shift to Greece when Rome is well-defended – it sounds interesting!