... whether somehow 'constraints' and genetic drift can funnel the progress of evolution towards complex ("designed-looking") characteristics that confer no adaptive advantage.

(To me, [that] still seems weird and unmotivated, though I'm not ideologically wedded to it being wrong.)

I suppose the epistemological point that Lewontin would harp on here is that if you go in expecting Nature to only do things that seem well-motivated and 'natural', then you are going to produce ideologically biased science.

But as to whether evolution can produce complex 'designed-looking' characteristics that are not positively adaptive, give a look to this single-topic blog by Arlin Stoltzfus and this paper discussing an idea known as Constructive Neutral Evolution (pdf).