Can someone summarize what motivated these seemingly arbitrary expressions?
They are far from arbitrary.
Because quadratic reciprocity is about whether a prime $p$ is a square modulo another prie $q$, it is natural to look for an actual square-root when it is (an idea that was already familiar to Euler). The use of quadratic Gauss sums is to construct the desired square-root (but possibly in a finite extension). With that explicit goal in mind, it is in fact not so hard to see patterns emerging when doing computations, especially if you have been thinking long and hard about Legendre's symbols and even if you don't know the answer beforehand. I have sometimes asked students to guess them for instance and they do succeed (occasionally).
But though the students do not know the result, they have a tremendous advantage: they are conducting the guessing in a course on finite extension of finite fields or algebraic number theory. That idea, i.e the idea of looking for the square-root in extensions (of the integers or of finite fields) and of studying the behavior of primes in such extensions, was absolutely groundbreaking.