When we did Gouraud shading didn't we interpolate illumination values that we found at each vertex using screen-space interpolation?

Didn't we just demonstrate that screen-space interpolation is wrong ?

Does that mean that Gouraud shading is wrong?

Yes, Yes, Yes

Gourand shading is wrong.

However, you usually will not notice because the transition in colors is very smooth
(And we don't know what the right color should be anyway, all we care about is a pretty picture).

There are some cases where the errors in Gourand shading become obvious.