In topology when studying a space with non-trivial fundamental group it becomes important to consider homology and cohomology with coefficients in representations of the fundamental group, i.e. local systems. The general Serre spectral sequence for a fibration with non-simply connected base provides an example. Another use of local systems is to get a version of Poincaré duality for non-orientable manifolds.
I am wondering about similar instances when it comes to e.g. algebraic de Rham cohomology, where the coefficient systems are given by quasi-coherent sheaves with integrable connections.
What does the ability to construct the de Rham complex with coefficients in a module with integrable connection buy us? Or for crystalline cohomology; what are some uses of the full theory of crystals (rather than just the obvious crystal given by the structure sheaf)?
(note the related question: de Rham cohomology and flat vector bundles was more focused on analytic and topological situation. Also, the question: Coefficients of Weil Cohomology Theories focused on conceptualizations of these coefficient systems, whereas I am more interested in applications.)
What are some of the uses of coefficient systems in algebraic geometry, especially in arithmetic contexts?
I guess one still needs these coefficient systems to prove duality statements?
I also think that Deligne's proof of the Weil conjectures relied heavily on using constructible $l$-adic sheaves. The same goes for Kedlaya's proof of the Weil conjectures using rigid cohomology, with F-isocrystals as coefficients. What are some reasons why one needs such coefficient systems in these proofs? What are other instances where this extra flexibility is needed?