Describing the physics in the interface between two different materials is a non-trivial task. Nonetheless, 'enormous amount of conducting material' is not a problem for so long as it does not undermine the assumptions underlying the 2DEG (2-dimensional electron gas) system; if such contact with the leads affects the uniformity and dimensionality of the electron gas on a macroscopic scale, then one has a problem, otherwise not.

Leads are very successfully described as reservoirs of charged carriers in thermal equilibrium, described in terms of a temperature and a chemical potential (that is, their voltage with respect to some ground voltage). When leads are connected to, say, a 2DEG,  in a current carrying state, where the 2DEG is not in thermodynamic equilibrium (it may be in a steady state, after the transient effects have died out), the question arises as to where in the leads the chemical potentials take on the respective bulk values. This question can be dealt with theoretically, using the non-equilibrium formalism. Ideally, if one has the necessary expertise and computing facilities, one can simulate both the 2DEG and the leads as one big system. In practice, this proves unnecessary, as professional experimental physicists are capable of experimentally verifying the validity of various assumptions underlying their measurements.