I try to answer here only to this part of the question: "does anyone out there actually use this definition to calculate a derivative that couldn't be obtained by a known symbolic rule?".

You wrote that using functions defined piecewise by different analytical expressions is no good for you, but I wouldn't be so categorical. This nice little educational paper by H.T. Kaptanoglu shows how far you can go using a function defined by a very simple closed-form expression everywhere except at one point: https://www.researchgate.net/publication/265989067_In_praise_of_yx_a_sin1_x

(I add that I completely agree, on the more general question concerning why to present the derivative that way, with the answer by Deane Yang.)