One thing that seems to be missing (and wasn't given a lot of attention when I was in school) was the notion that the reals are continuous. Calculus made a lot more sense to me once I internalized how that one simple idea basically serves as the rug that really ties the room together.
I've often thought that an interesting treatment would start with differences and sums of integers as approximations, demonstrate their errors and then introduce reals and limits as a tool for making better theoretical models using the infinite "zoom button" continuity property of the reals.
You may want to check out Concrete Mathematics if you haven't already. "Concrete" is a punny sort of blending of "Continuous" and "Discrete" and it was while working through this text that I had a lot of calculus revelations that would've helped me out in my first couple years of college.
I've often thought that an interesting treatment would start with differences and sums of integers as approximations, demonstrate their errors and then introduce reals and limits as a tool for making better theoretical models using the infinite "zoom button" continuity property of the reals.