As an example of a major doing things correctly, in my opinion, is MIT's aerospace engineering with their "unified engiineering" course that covers the basics in a highly integrated fashion.
My experience in a different engineering program was that I learned some sophomore math freshman year (because I had placed out of "freshman calculus"), then didn't use it at all until junior year when I had forgotten it. And then things tried to be integrated senior year when we had a senior design project that tied all our random classes together. I turned out alright, but I think an integrated approach would be much more fruitful.
All that said, I do think there is some value in having the purely mathematical ability to look at an equation, classify it as a certain type, and then know how to solve it. Not every problem you face in life will be an application that existed when you were in college.