One repeatedly hears complaints about the irrelevance of certain courses in an academic program. That the course added no value; and besides, one never will apply that knowledge to one's life, and so why bother.
Why, indeed. One purpose of education is to learn "facts". Facts are important, of course. It is important to know that there are ten millimetres in a centimetre. And a hundred of those in a metre. That the earth is approximately spherical and goes around the sun; and who is the current leader of the nation, and so on. But education has an even more important purpose. One thing that everyone will require to do throughout his or her life is to understand issues and solve problems. Education offers, above all, models, and theories. Models, in abstraction are metaphors, and metaphors are essential to understanding. For everything, ultimately is understood in terms of something else. The larger and more varied one's cache of metaphors, the better one can understand the world in all its complexity, and hence more effectively solve problems. You may never need to construct a chemical process flow involving osmosis, but you almost certainly will find many opportunities to apply the metaphor of osmosis to numerous situations in life.
Yes, a broad education does make you much smarter than a narrow one, because of the rich variety of metaphors it grants you.