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.
It was so interesting that the professor would do theorems walkthrough classes which were optional and everyone would attend them.