The German government is the shareholder and appoints the board, but Deutsche Bahn is otherwise operated as a private company. For example, it operates at a profit, its CEO is compensated comparably to other corporate executives, and it competes with other rail operators both inside Germany (it has subsidiaries that compete to operate regional train service), and internationally. Amtrak is operated, for all intents and purposes, like a government agency. It operates at a loss, its CEO is compensated like a government officer (no incentive to cut costs to boost bonuses), Congress exerts significant political input into operational decisions, it does not compete with anyone inside or outside the U.S.

Germany also has many fewer freight trains than the US. Their rail network is optimized for passenger use, and it works well.

The US rail network is optimized for freight hauling (and it's largely owned by freight railroads and maintained at private expense, and those railroads turn a consistent profit). It also works well -- for its intended purpose.

If you want to move 3000 miles at 45 MPH remarkably cheaply, and you happen to be (or live in) a cargo container, the US rail network is built just for you!

Competition is a bit of a red herring - it's not as if anybody is stopping anybody from starting their own train service in America tomorrow. The reason Amtrak is run like a government agency is because if they didn't offer routes at a loss, nobody would run those routes at all.