As I stated in the comments the post by Arnold Neumaier in reply to this question answer mine too. In particular the papers he links (arXiv papers cond-mat/0102428 and cond-mat/0203460). In my mind the two papers are essentially a continuation of what Leggett and Caldeira showed (that the density matrix becomes diagonal when coupling to certain statistical ensembles). Namely Nieuwenhuizen et al. demonstrate by solving a representative system analytically that the complete measurement process (or "collapse of the wavefunction") can be seen as such a statistical coupling. From the abstract to the first paper:
A model of quantum measurement is proposed, which aims to describe statistical mechanical aspects of this phenomenon, starting from a purely Hamiltonian formulation. The macroscopic measurement apparatus is modeled as an ideal Bose gas, the order parameter of which, that is, the amplitude of the condensate, is the pointer variable. It is shown that properties of irreversibility and ergodicity breaking, which are inherent in the model apparatus, ensure the appearance of definite results of the measurement, and provide a dynamical realization of wave-function reduction or collapse. The measurement process takes place in two steps: First, the reduction of the state of the tested system occurs over a time of order ℏ/(TN1/4), where T is the temperature of the apparatus, and N is the number of its degrees of freedom. This decoherence process is governed by the apparatus-system interaction. During the second step classical correlations are established between the apparatus and the tested system over the much longer time-scale of equilibration of the apparatus. The influence of the parameters of the model on non-ideality of the measurement is discussed. Schrödinger kittens, EPR setups and information transfer are analyzed.
For me this answers what I was trying to get at (though didn't really succeed at formulating) in my question: that just claiming a statistical process "does the job" for collapsing the wavefunction is not enough, there are non-trivial features of the collapse that have to be explained and modeled appropriately. In the two papers this is done for analytically solvable systems.