Almost every country taxes people who live there. The United States taxes people who are American, wherever they live. Moving to Italy does not switch that off, does not pause it, and does not reduce the filing obligation.
So the American arriving in Italy is not choosing between two tax systems. They are living inside both, simultaneously, for as long as they hold the passport. Everything then depends on how the two interlock: which country has the first claim on each type of income, what credit the other gives for it, and where the machinery simply fails to mesh.
The Italy-US double tax treaty manages much of this. But it contains a saving clause which expressly preserves the right of the United States to tax its own citizens as if the treaty did not exist. Most treaty benefits you might expect to shelter you, as an American, are switched off by that single provision.