Neil,
I have a rather different take on the Born Rule:
Wave-functions are described mathematically in terms of Fourier Transforms. The transforms are invariably mis-interpreted as physical (rather than purely mathematical) superpositions. However, Fourier Transforms have another, very different, physical interpretation, commonly used in communications theory; they are tuned filter-banks, in which the "power spectrum" measures the power received within each narrow "bin" of the transform. If the input signal consists of a set of identical particles or wavelets, each carrying an identical amount (quantum) of energy, then the power spectrum is it fact, nothing more than a histogram, in which the total energy received in each bin, divided by the energy per particle, equals the number of particles received per bin. That is why the whole process corresponds to a probability measurement - it literally is - a histogram.