Another nice reference that compares different approaches to proving Sobolev inequalities is Thierry Aubin's "Some Nonlinear Problems in Riemannian Geometry". He gives three arguments: the ones using HLS and Gagliardo-Nirenberg, and a third one due to Talenti and himself which not only shows that the Sobolev inequality is essentially equivalent to the isoperimetric inequality, but also gives the best constant.

I've repeated the steps of your argument and I think I've got it now. The point of your line integration along the second argument was just to display the additional conormal singularities of the kernel at boundary points, correct? It did take me some time to visualize this... Thank you very much for the patience and help!