I believe it gives a good conceptual or practical reasoning to why we would want to study calculus in the first place. As it was first introduced to me, we can always talk about the average speed a car has traveled over a certain distance. Even very small distances, but apart from a speedometer, how can we say 'I'm travelling XX km/h right now'. There enters the limit definition, where we want the instantaneous rate of change!

If we only presented the formal rules for differentiation, we run in to the same problem as high school students who dislike math present "But my calculator can just do it! Why do I need to learn this?!". If the fundamentals are not taught, one day they will be forgotten.

There are certainly other rigorous approaches to the derivative out there. The delta-epsilon method, which most students in their first year simply struggle to grasp as easily as the $h \rightarrow 0$ definition. This approach is typically reserved for the math majors who go on to take a course in analysis, not the general first calculus course for all science majors.

While I do not use this definition in practice, I am primarily not calculating derivatives, so take that for what it's worth I suppose.