I seem to have seen a way of dealing with connected masses in Resnick and
Halliday. Here you look at the tangential forces acting on the system of
strings and blocks. Forces perpendicular to motion direction balance out for
each block. You could then include the connected blocks and the strings in
the system and look at the tangential forces alone. The problem of Atwood's
machine would reduce to saying that force on one block is along a positive
tangent and on the other along a negative tangent. The rope as it passes
around the pulley experiences normal forces due to the pulley, so they do
not count.