So a naïve proposition; what if I view my ANN weights as a probability of being a 1 or 0; not as a conductance?  For each weight, I just stick my weighted coin in there, and I get a nice efficient binary weight matrix that I can run many times to collect a lot of independent samples. 4/

Well, for a single run of the network, that random binarized network is really noisy and any single output is pretty likely to be wrong.  But if you do this many times, those samples create a distribution which… amazingly… looks like a distribution you would want from a classifier! 5/