Linear systems lend themselves to analysis since they follow certain rules. Because of this, much analysis has been done and theory developed, so there is now lots to keep undergrads busy with.
Many simple passive systems are linear systems, or close enough so that they can be approximated as such with useful result. All that theory can be applied to understanding what they will do with specific inputs, and figure out what inputs to give them to yield particular outputs.
This theory together with Fourier analysis can be quite useful. You can talk about things like the frequency response, step response, impulse response, and tap into the existing knowledge how to design filters, know how close to instability a control system is, how many points and what values to use for a convolution filter kernel, etc.
Without linear systems theory, you'd be making it up from scratch in each instance.