Gram-Schmidt takes a perfectly good basis and replaces it with an orthonormal one. Why is that worth the computation, when the original basis already gives every vector unique coordinates?

A. Because each coordinate becomes a single inner product.

Both bases give unique coordinates and differ entirely in what it costs to get them: a general basis means solving a linear system for every vector, where an orthonormal one gives cᵢ = ⟨w, eᵢ⟩ because every other term is killed. That independence is what makes the idea far-reaching. Fourier series are this construction in a space of functions.