When I was doing an associate's in engineering, our calculus and differential equations courses were like this. We'd learn some math, do some problems by hand, then we'd have a lab component where we were introduced to either methods in a computer algebra system or some numerical methods. The problems we solved there were word problems that had the higher level physics already set up for us, so that we ended up just having to solve the calculus or differential equation portion of the problem.

The calculus books we used were not set up like this and the books that focused on learning the CAS or numerical methods weren't structured any better. I think this only worked because it was a small program aimed at technical education with a faculty that cared about developing a unified curriculum.

When I transferred to a different university to finish a degree as a stats major, all of our courses and most of the textbooks were structured in a way to use R. We did some problems on simple linear regression by hand, but very quickly it becomes impractical do to it any other way. This seemed very natural to me, but apparently it was not the typical experience of studying statistics.

Perhaps there are some calculus books out there that do a good job of both teaching calculus concepts and using CAS / numerical methods, but my narrow minded view is that calculus is a tool for physics, engineering, or other applications, and you'll be bogged down in teaching the relevant domain knowledge to get interesting examples. If you're looking for your own examples, perhaps this could be done purely through the differential calculus topics of related rates and optimization or the integral calculus topics of simple ordinary differential equations.