This is a classic debate in the philosophy science. Should theories describe the world, or merely make predictions about it? Because no one these days believes that quantum mechanics really describes the world, we live firmly in the realm of theories being things that make predictions.

The existence of the atom, however, is not what is being predicted. A theory should predict the sensor readings of particular instruments during particular experiments. Those sensor readings are what is real; "The Atom" is the abstraction.

The very idea that "things exist" is an abstraction. A thing exists if we can model it, and that model corresponds with reality.