In a typical class on single variable calculus, I would imagine that you could get away without using the $\varepsilon-\delta$ definition directly and just using things like L'Hopital's rule. Certainly, when calculus was first formulated, it was done so without that rigor, and yet people were still able to get correct (and even astonishing) results. However, this is more a consequence of most familiar examples falling under a few rules than a case where a few rules cover everything imaginable.
That said, when we try to avoid the definition, we end up needing a lot of rules. For instance, if we want to know that
converges, we want the alternating series test, but if we want to know that
converges, we want the integral test. Of course we can prove all these things and never worry about $\varepsilon-\delta$ again, but we end up assuming a lot of machinery which we had to construct.
In particular, when presented with a new kind of problem, we might need to go all the way back to $\varepsilon-\delta$ in order to build more machinery. For instance, consider the function:
which can reasonably be seen to converge. We might wonder if $f'(x)$ exists anywhere. Termwise differentiation gives
which doesn't converge anywhere (since the limit doesn't go to $0$). Problem is that the termwise derivative not converging doesn't necessarily tell us anything about the function itself - for instance, we easily have that
because the sum telescopes, but differentiating both sides (assuming that termwise differentiation works) gives:
where the terms on the left go to $\infty$, so the sum doesn't converge. So, thus, we're left without tools other than $\varepsilon-\delta$ to approach such a problem (and it turns out not to be trivial).
My point here is that, yes, we usually abstract away from the $\varepsilon-\delta$ definition. We try to prove lemmas which capture our intuition and which allows us to work intuitively, yet support our work rigorously (without any mathematical contortion beyond the initial proof of whatever lemma we wanted). And, indeed, if we have an intuitive answer, we probably have some "higher level" theorem we can apply to get that result to work well with our intuition. However, given that $\varepsilon-\delta$ defines the central idea of study, sometimes we need to go back to the definition either to prove more lemmas, or to investigate some object which defies our usual approach.