If you believe that the monomorphism $\pi \to \widetilde{\widetilde{\pi}}$ is an isomorphism for admissible $\pi$, then you can see that $\pi \mapsto \widetilde{\pi}$ is a contravariant autoequivalence on the (abelian) category of smooth irreducible representations. Thus it carries simple objects to simple objects.

If you believe that the monomorphism $\pi \to \widetilde{\widetilde{\pi}}$ is an isomorphism for admissible $\pi$, then you can see that $\pi \mapsto \widetilde{\pi}$ is a contravariant autoequivalence on the abelian category of admissible (smooth) representations. Thus it carries simple objects to simple objects.