Substitution fails on $\int \cos(x^{2})\,dx$ because the derivative of the inner function is not present as a factor.

True

True. Substitution undoes the chain rule, so it needs the inner function's derivative sitting there as a factor, the $2x$ that $\int 2x\cos(x^{2})dx$ has and $\int\cos(x^{2})dx$ does not. The second has no elementary antiderivative at all; the Fresnel integral is defined to name it.