I'm going to answer this part:

does anyone out there actually use this definition to calculate a derivative that couldn't be obtained by a known symbolic rule?

Yes. $sin(x)$.

My point is that of course we can just learn the derivative of this function, but then we could just learn the derivative of any function. So looking for a "complicated function" that needs the limit definition is pointless: we could just extend our list of examples to include this function. It's a bit like the complaint that there's no closed form for a generic elliptic integral: all we really mean is that we haven't given it a name yet.

In fact, one could do $x^2$ like this, or even $x$, but I think that $sin(x)$ has a good pedagogical value. If you can get them first to ponder the question, "What is $sin(x)$?" then it might work. I'm teaching a course at the moment where I'm trying to get my students out of the "black box" mentality and start thinking about how one builds those black boxes in the first place. One of my starting points was "What is $sin(x)$?". Or more precisely, "What is $sin(1)$?". If you take that question, it can lead you to all sorts of interesting places: polynomial approximation of continuous functions, for example, and thence to Weierstrass' approximation theorem.

Many students will just want the rules. But if the students refuse to learn, that's their problem. My job is to provide them with an environment in which they can learn. Of course, I should ensure that what they are trying to learn is within their grasp, but they have to choose to grasp it. So I'm not going to give them a full exposition on the deep issues involving the ZF axioms if all I want is for them to have a vague idea of a "set" and a "function", but I am going to ensure that what I say is true (or at the least is clearly flagged as a convenient lie).

Here's a quote from Picasso (of all people) on teaching:

So how do you go about teaching them something new? By mixing what they know with what they don't know. Then, when they see vaguely in their fog something they recognise, they think, "Ah, I know that." And it's just one more step to, "Ah, I know the whole thing.". And their mind thrusts forward into the unknown and they begin to recognise what they didn't know before and they increase their powers of understanding.

We all remember professors who forgot to mix the new in with the old and presented the new as completely new. We must also avoid the other extreme: that of not mixing in any new things and simply presenting the old with a new gloss of paint.