One thing you left out, your engineering courses took real effort and the subject matter required problem solving and math. So we see how simple it is to select a major that will lead to employment.
Become a problem solver, understand the math to the level that you future employment uses. Where do I teach? Dept of engineering and technology.
Obviously, you should feel good, and do feel good about what you are doing to help young people to become employable and in helping them to develop and start a career.
Imagine how the fake department professors must feel after bamboozling their young students into thinking they are studying something that will help them later in life!