How to motivate the axioms for the inner product

Symmetry, bilinearity, and positive definiteness are exactly the properties used to prove the Cauchy-Schwarz inequality. Well, there are a zillion proofs of the Cauchy-Schwarz inequality; I mean the one that proceeds by observing $0\le \|x-ty\|^2$ for all $t\in\mathbb R$, expanding to obtain a quadratic in $t$, and concluding that the discriminant of that quadratic is nonpositive (and then you fiddle with definiteness to get the equality case).

In other words, an inner product is just a map for which that proof is correct.

We want to obtain the Cauchy-Schwarz inequality in other spaces because it's a cornerstone of the linear-algebraic treatment of Euclidean geometry — you use it to prove the triangle inequality, to show that orthogonal projections are metric projections (which gets you everything you want to know about tangent planes to spheres), etc. (The equation (1) is part of all that: in this treatment, it's essentially the definition of angle. You need Cauchy-Schwarz to show that it's well-defined.)