In their original paper the authors derive a phase between the
acceleration of the source and the advanced wave at the source coming
from the absorber. This phase is needed in order to conclude that the
reaction is proportional to the time derivative of the acceleration
rather than the acceleration itself.

To arrive at this phase however the authors argue that while the speed
of the retarded wave is reduced by interacting with intervening medium
between the source and the absorber particle in question, the speed of
the advanced wave on its way from the absorber particle to the source is
the speed of light and does not interact with intervening particles. The
reason for this they say is that unlike the retarded wave which they
treat as the combination of the primary retarded wave and the secondary
waves that come about from the interactions with intervening particles,
the advanced wave is an ELEMENTARY interaction between the absorber
particle in question and the source and therefore does not interact with
other particles. What is this difference between the treatment of the
advanced and retarded wave motivated by?

Why is one considered
elementary and the other not?

Isn't this a time asymmetric assumption?

If the answer to the last question is yes then why do we need to blame
the time asymmetric dominance of diverging waves over converging waves
on asymmetric initial conditions why not also blame it on this
asymmetric treatment of advanced and retarded waves?