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You are correct. One can start at either end of the chain and arrive
This is a "straw dog" argument. Momentum conservation comes quite
naturally from applying Newton's Laws to collisions. N3 gives the
forces as equal and opposite, the collision times are the same,
N2 gives the momentum changes of the colliding objects to be
equal and opposite. Momentum conservation becomes a simple
consequence of Newton's Laws - not an ad hoc quantity invented
for the purpose of rationalizing data. Starting from momentum and
then deriving Newton's Laws is certainly equivalent, but I don't
see how it is superior. Both approaches require careful study of
various simple physical interactions, each can then be used to
produce the other, and they both have the same problem with the
definition of mass.