In my experience, epsilon delta understanding was pretty much unrelated to passing calculus, where tests consisted of formula/substitution crunching.
That's a shame. There are so many exciting things to learn in calculus that you can skip the epsilon delta stuff and still do so much more than formula/substitution crunching. Calculus is the gateway to differential geometry, topology, mathematical physics, differential equations, taylor series which are useful for numerical approximations and so many other things, analytic number theory, complex analysis, many forms of statistics. So many wonderful things you can do with it.
Imagine using your new found calculus knowledge to show that the orbit of the planets about the sun is an ellipse -- which is what Newton used calculus to do -- OR learning all about epsilons and deltas. Which would be more fun as a student learning calculus. Make the math exciting enough and people will put up with the drudgery of calculations.
e.g., many engineering calculus courses, and explicitly some of the texts.