So, yes, the “Adding Probabilities” method is wrong. As it turns out, the reason I was getting such bad distributions when using Mathematica to produce graphs for the (correct) “Adding Fields” method is that I had not properly adjusted the phase for each field point source within each of the slits. When I do that, it produces distributions in the near- and far-fields that, I think, are consistent with what would be observed in actual experiments.
But this essay is important to me for several reasons. First, it underscores one of the problematic assumptions I’d been making, namely the assumption that there is some reality about where, in each slit, a particle is located. Identifying that as incorrect helped me come to what I believe is a better understanding/interpretation of QM, which I describe here, in which a superposition is indicative of a lack of a fact. Second, writing it helped me to understand the relationship between single-slit Fraunhofer distributions and double-slit interference distributions. Third, it makes some good points about problems in QM, and is mostly correct if you’ll ignore any nonphysical wackiness in the “Adding Fields” graphs.
I have made huge progress in understanding physics over the past couple years and wouldn’t be where I am today without the experiences of yesterday.