I think you overstate the non-analog nature of a neuron. Firstly, a neuron rarely fires only once, and the times between spikes are analog.
Secondly, the state of a neuron that determines whether it fires is its charge, which is determined by the number of different ions inside it, so is like a low-precision float.
Finally, a neuron gradually discharges all the time, so the charge is not simply proportional to the count of incoming spikes.
In any case the nature of the basic processing unit doesn’t appear to matter as long as there’s some nonlinearity involved — people even managed to make a NN work by using the float rounding behaviour around zero as the nonlinearity ☺