Nut makes a difference when fretting because the tensile wave still travels past your finger, and is damped, reflected or passes through the nut depending on frequency, and the nut properties effect where those different frequencies are. Plus the harder nut retains its shape better, so a correctly shaped nut will remain correctly shaped. You can demonstrate the importance of the string below the fretting point to tone but fretting a not and playing it, and still fretting it and cutting the string below the fretting point, then comparing tone, the difference is certainly audible.