The deeper reason for why the position representation in non-relativistic QM is so useful is the fundamental structure of this particular realization of a QT, which is governed to a large extent by the underlying symmetry principles, most importantly of space-time. In this case it's the Galilei group, which is a bit tricky and from a mathematical point of view even more complicated than the Poincare group underlying special-relativistic spacetime. The analysis of the ray representations of the Galilei group or, (nearly) equivalently, the unitary representation of the central extensions of its covering group (which adds to the classical Galilei Lie algebra the mass as a non-trivial central charge and substitutes the rotation group SO(3) by its covering group SU(2)), leads to the typical structure of the Hamiltonian in terms of kinetic energy plus a potential that is only a function of the position variables, where one should be aware that the position variables are constructed from the representations of the quantum Galilei group for ##m \neq 0##; the case ##m=0##, i.e., the unitary representations of the classical Galilei group do not lead to useful quantum dynamics (Wigner and Inönü).
In the relativistic case the Poincare group has no non-trivial central extensions, and the mass squared is a Casimir operator of its Lie algebra. So far only the cases ##m^2 \geq 0## lead to successful models for physics. For ##m^2>0## one can always construct a position observable from the Lie algebra of the (positive orthochronous) Poincare group, but in the relativistic regime particles are not as localizable (i.e., it's position cannot be determined to be in a small volume) that easily. Losely speaking in an attempt to squeeze the particles into a small volume you need pretty high energies, which rather leads to the creation of new particles than a more accurate localization of the present ones.
For massless quanta of spin ##\geq 1## a position observable cannot even be fully defined. As with classical light (em. waves) photons can only be localized in some sense in transverse direction but not in beam direction. In lack of a proper definition of a position observable, what's meant by "localization" is rather a detection probability distribution that is narrowly peaked in some volume. The detection probability distributions are essentially what defines the observable facts about photons. In the most adequate formalism, which is QED, it's well defined by adequate correlation functions of field operators.