I guess I have an engineering viewpoint as well--which may not be surprising because I, well, was one. I never especially connected to geometry with its proofs in high school. What I did learn was lots of basic trig and algebra and the beginnings of calculus (which I took more of in college). The result is that while I've never had much of an interest or flair for advanced math, I did develop most of the math skills I needed both for school and, more importantly, through career and life.
I've had stats as well. As I was just discussing with someone in a different context, I think stats, accounting, and similar applied math would be far more useful to most students than either most calculus (past simple differentiation which is used in a lot of contexts), proofs, or set/group theory. And, arguably, could be linked a lot better to real-world applications.
The question is what are you trying to develop in your students: factual knowledge or mathematical thinking skills? I think it's the latter, and if that's the case then proofs are by far the best way to do it.
And do you really think students would be any more motivated to learn about accounting than they are about current math? I mean, come on, nobody wants to do that stuff. It's a chore even for adults.
I've had stats as well. As I was just discussing with someone in a different context, I think stats, accounting, and similar applied math would be far more useful to most students than either most calculus (past simple differentiation which is used in a lot of contexts), proofs, or set/group theory. And, arguably, could be linked a lot better to real-world applications.