My quick response on completely inelastic collisions being where the
objects stick together after the collision was at the beginning
physics level such as I do in both our calculus-based and
algebra-based two-semester sequence of courses. In these courses the
class has not been introduced to anything other than straight-forward
mechanical forces (those from weight, friction, normal force,
springs, and otherwise pulling or pushing). So there is no reason to
include long-distance forces and their complications, although one
should mention them as an aside to remind students that there is
"more to come."
My presumption was that this was the level of the question that began
this thread. In fact the usual general physics books, such as Serway
and Jewett and others, do not introduce center of mass coordinate
systems until after collisions are examined. To me this makes some
sense since our intuitive feel is for the frame of reference in we as
the observer are located. I do introduce center of mass coordinate
systems extensively in upper level courses, such as modern physics, as needed.
Just a more extensive rationale for my earlier terse response.