The point isn't what you use the FFT for but what the FFT does. The FFT does fast polynomial multiplication.
Look at a system doing DSP for example. Let's say you have an input RF signal that goes into an analog band-pass filter, a mixer, an analog low-pass filter, a DAC, and a FPGA. The FPGA implements a FIR filter - lets say a Hilbert transform using a distributed arithmetic algorithm. You are getting envelope information out of the FPGA. There's no FFT anywhere in the FPGA to compute this filter output.
You are seeing some weird noise in your DSP output. You dig a little by looking at the output of the DAC and then look at the FFT magnitude log10. You see some suspicious spurs popping up, but at really strange discrete frequencies. You show the spectrum to an RF engineer, then you two go to the lab and put an analog spectrum analyzer on the input to the DAC. Sure enough, there are mixing products of your input carrier, mixer frequency, and sampling clock frequency. The weird spur frequencies make sense now - you are seeing folding of the mixing products from the sampling. You tweak the sampling clock to move the spur digital frequencies.
In that long but real world example, I don't see anywhere here where thinking of things in terms of polynomials would be helpful here. It's also going to lead to confusion of terminology with your RF engineer - they tend to think continuous.
First, you learn about the convolution operation that, given any input function x(t), get the output of any linear time-invariant circuit as y(t). The convolution integral is nasty, professors make you feel the pain a bit. Then introduce the Laplace transform to make the computation much easier. Then go to continuous time, continuous frequency Fourier transform. Talk about frequency domain a bunch, learn filter topologies, etc. Circuits class over.
Now comes a signal processing class where you learn about discrete-time signals. First, talk lots about sampling. Then, get introduced discrete time convolution. Now, learn the z-transform and the discrete time Fourier transform (DTFT) for a discrete-time, continuous frequency signal. Mostly discuss filtering and spectral analysis. Intro to signal processing class over.
Learn about sampling the DTFT in the frequency domain. This is the DFT, which is usually presented as a sum that would require O(n^2) operations to compute. Learn that this corresponds to circular convolution and learn about zero padding for traditional convolution. Finally, get presented with Cooley-Tukey FFT algorithm for base-2. Focus is still signal and spectral analysis. Talk lots about windowing. You may get a mention that convolution corresponds to polynomial multiplication here. Or maybe they talk about grade-school multiplication, its really the same thing as polynomial multiplication with a carry. Senior level signal processing class over.