> If you write simple tests for those it is a type checker on it's own and the type checker is just overhead.

This is not true. Unit tests cannot replace a type system, just like a type system cannot replace unit tests.

You need a unit tests because a type system cannot check for all possible types of correctness.

However, unit tests can only check for the presence of bugs; they cannot prove their absence. On the other hand, a type system can prove that certain classes of errors cannot exist in the program.