Although indeed reciprocal space can be considered primarily a useful mathematical tool, I would not say that it does not "exist". For example, if you write down the description of a plane wave in terms of e^(irk) you would certainly not say that the position vector r exists but the k vector doesn't.. Using the de Broglie relation you can also interpret k in terms of momentum and certainly you would not consider position more physically real than momentum. In quantum mechanics for example it is your choice to represent the wave function as Psi(r) or via a Fourier transformation as Psi(k). In the language of quantum mechanics you would say you just changed the basis. Similarly, in geometry of curved spaces people introduce covariant and contravariant coordinates. There is no difference in Euklidean geometry, but as soon as you leave Euklidean space the same vector has very different representation in the two dual basis sets. No one would say that one should be favored over the other.

In my experience, the most natural way to introduce reciprocal space is via elastic diffraction physics. That can be X-ray diffraction or neutron diffraction etc. Perhaps neutron diffraction is even the simplest case. You consider an incoming neutron as a plane wave characterized by a wave-vector q. The neutron is elastically scattered by a target such that only its direction is changed. That means the outgoing plane wave has a wave-vector of identical magnitude but different direction. The change in direction means nothing other than a momentum transfer between neutron and target. Therefore you measure neuron intensity as a function of delta_q which spans your reciprocal space. The neutron intensity I(q) is related to the density of matter rho(r) in real space, because the neutrons scatter more the more nuclei there are in a given volume element of the sample. The relation is again given via a Fourier transform. The intensity you measure is the absolute square of the Fourier transform of the matter density. The essential phase factor of the Fourier transform is e^(ir(delta_q)) obviously a direct result of the change of the incoming plane wave into an outgoing plane wave with different direction. All elastic scattering follows the same principle whether it is neutrons, electrons, X-rays or light. Therefore the universal importance of reciprocal space. Likewise it is straightforward and much related to introduce the reciprocal lattice in the context of diffraction physics. The reciprocal lattice appears when you Fourier expand a density rho(r) which has periodicity in real space as you have it in crystalline solids. That explains the importance of reciprocal space and reciprocal lattice in condensed matter physics of crystalline solids. Whenever you have periodicity you will find reciprocal space description, and the reciprocal lattice playing an important role. For example Bragg's law of X-ray scattering means nothing but that the scattering vector (the difference between incoming and outgoing wave-vector) has to be a reciprocal lattice vector for constructive interference condition. Also the concept of the Brillouin zone is related to the reciprocal lattice. The first Brillouin zone is nothing but the Wigner Seitz cell of the reciprocal lattice.

In quantum mechanics the wave functions |psi> can be represented in different ways. One way the the real space representation of the wave function psi(r) = , another possibility is the k-space representation psi(k) = . Both representations are conneted via a Fourier-transfomation. In this sense both representations exist.