It's not only the Born's rule. All the equations have a similar symmetry. (You may have ∂Ψ=VΨ, or Ψ∂Ψ*+∂ΨΨ*+ΨΨ*. See the real examples in the link in Wikipedia. But you never have something like ΨΨ+Ψ*Ψ* that mix the number of times that the linear and the conjugate version appears.)

Also, it's not so immediate, because you must be stubborn enough to think that a global obvious symmetry "must" be extended to a local symmetry. And in any case, it took like 40 years a few brilliant persons to discover it.