> Even puny single-precision floating point operations are accurate to something like 7 decimal digits and have a dynamic range of hundreds of dB. According to the spec sheet, the appliance only uses 6-bit inputs and 8-bit outputs, so the relative errors are probably on the percent level. This makes it hard to believe that any signal will propagate through something like a DNN without completely drowning in noise.

Maybe you aren't aware, but half-precision (16 bit float) is already well-established in the AI community: https://en.wikipedia.org/wiki/Bfloat16_floating-point_format. In context single-precision isn't all that puny!

And there have already been successful experiments with stronger quantization, like 8-bit neural nets, or even 1-bit (!) neural nets. There is a lot of evidence that neural networks can be very resilient to quantization noise.