To the OP,
Your best chance of employment after finishing training is in Europe. Not necessarily Italy, but definitely somewhere in the EU. I have many friends who came to the US for flight training and wanted to stay to work, and the overwhelming majority are back in Europe. There are some that managed to stay in the US beyond their OPT experience, but a couple won the green card lottery and the rest found love in the US and got married. If you want to come to the US to train, you need to be fully prepared to go back to Europe and do an EASA license conversion and work there once the time limits reach their end.
I can’t comment on Australia, but my company has been hiring many Australians with an E3 visa and when I ask them about the pilot situation there, it sounds like many are coming here to help launch their career.