This stuck out to me, and might be a place where our intuition breaks down. As a human if you tell me, "give me a polynomial with degree n with coefficients drawn uniformly from [-1, 1]," I will likely produce one with a lot of zero coefficients, just because a lot of the polynomials you encounter in the wild have a lot of zero coefficients.

In reality, if you grab a coefficient uniformly from [-1,1], the probability that it is 0 is 0.