Please help inform this ignorant overprivileged Australian, but do workers in the US not have the right to join a union?
Yes, but many states are 'right-to-work' states, which means that you don't have to join the union to work in a given field. And most states are 'at-will' states which means that an employer doesn't need a reason to lay off workers, and there was a recent Supreme Court decision of Janus v. AFSCME that forbid the automatic gathering of union fees for public section unions.
Due to a number of factors (corporation opposition, conservative politicians opposition, globalization, etc.) union participation have steadily declined for decades.