(In this question, $(*)$ means normal when working over $\mathbb C$ and means self-adjoint when working over $\mathbb R$.)
This question is related but despite the same title what that question actually asks is what is the importance of diagonalizing a matrix. I think we can all agree that diagonalization of matrices is useful. Is there anything more to the spectral theorem than knowing when a linear operator can be diagonalized with respect to some orthonormal basis?
To further motivate this question, here's a quote from the book Linear algebra done right:
The Spectral Theorem is probably the most useful tool in the study of operators on inner product spaces.
It certainly is useful for $(*)$ operators. But is it useful to study operators that aren't $(*)$? It would seem weird to call it the "most useful tool" in this area if it is only useful for this kind of operators, which are in a sense the "simplest" ones.
An answer to this question could either provide intuition for the importance of the Spectral Theorem or give examples of applications of it to non-$(*)$ operators. Answers (only) about the importance of diagonalizing matrices are obviously not welcome since we already agreed on the usefulness of that.
The Spectral Theorem leads to the Singular Value Decomposition and to the Polar Decomposition, both of which are hugely important and apply to all operators, not just self-adjoint or normal operators.