> Half the time, a bit sampled from the first input is transferred to the output; otherwise a bit from the second input is used, effectively averaging the two inputs. The circuit that accomplishes this sampling is again a very rudimentary one, called a multiplexer. With it, addition becomes very easy.

A multiplexer with a magic random input? If you merely alternate, then you have a non-i.i.d. sequence, and your next gate can mess up.