The intuition and the formalism are presented together (at least, they should be!). To learn the role of epsilon and delta, the student needs to jump back and forth, finding the correspondences between equations and the motivation. This is a skill that needs practice; this was one of the first places I found the equations dense enough that I couldn't just "swallow them whole".
(The earlier I remember is the quadratic formula, which I first painfully memorized as technical trivia. It took me a couple of years to grasp that it was completing-the-square in general form. Switching between the general and the specific is another skill that you develop)
https://en.wikipedia.org/wiki/Elementary_Calculus:_An_Infini...
When I was in college, the Philosophy department offered this course. It was considered an easy way to get a general education math credit without needing to be good at math. It was a really enjoyable course[0] that put me on the path to becoming a computer programmer. It occasionally comes in handy[1].
If you mean experimental work, then sure, that's like laboratory chemistry. You run code and write up what you observe happens. If you're trying to prove theorems, you have to understand the epsilon delta stuff even if your proofs don't actually use it. It can be somewhat abstracted away by the statistics and differential equations theorems that you mention, but it is still there. Anyway, the difficulty melts away once you have seen enough math to deal with the statistics, differential equations, have some grasp of high dimensional geometry, etc. It's all part of "how to think mathematically" rather than some particular weird device that one studies and forgets.