The center of mass can be used as representing a whole system only
when the interactions considered are the same (up to approximations)
for all points of the system, or average linearly with respect to
distance, like the computation of the center of mass itself.
This is not so for gravitation, because of the inverse square law of
attraction. The center of gravitation of earth is 6,300 km from us, and
it is 400,000 km for the moon. So the moon is 63 times further away,
and its gravitational effect on objects on the surface of Earth is then 4,000 times weaker for the same amount of
mass. It does have an influence, seen as tides. But we are in no way
atracted towards the center of mass of the moon-earth system.
Note that I talked of center of gravitation of earth, because, if I
remember correctly, a homogenous sphere, or spherical shell, attracts
towards its center. You were quite right on this.
To answer your next question (which you should delete and edit into the first):
(sorry if this is a bit messy, the question, from a very new user, has been evolving a bit too much)
The center of gravity could be defined as the point with respect to
which the torque of external gravitational forces applied to a rigid system
averages to 0 (I do not know whether that is actually done). It is the
same as the center of mass if the system is in a uniform gravity
field. Actually the expression was created for objects in a uniform
gravitaty field. It allows you to compute trajectories, independently
ot rotations. It is often loosely used in place of center of mass.
Except for spheres, it is not the point of gravitational attraction of a
system. As we have just shown, such a point does not exists in
general. If you consider Earth-Moon as a single objet, you are
attracted to the Moon when close to it, and to the Earth when close to
it. In general the gravity vectors do not converge through a unique point.
I think you should just forget center of gravity in non uniform
gravity fields. Regarding dark matter, I believe astrophisicists look
at observable mass, and analyze motion with respect to the observed
mass. Given that observed motion does not follow the usual laws, they infere
that it could be because there is some invisible mass that contributes to
the gravity field. It could be that the invisible mass does not have
the same center of mass (visible and invisible
mass attract each other, but they could be rotating around a common
center of mass). But it also has an effect on the speed of rotation of
visible bodies. So, if a proper distribution of such an invisible mass
can explain observed motions, that is a simple explanation for it and
thus it has a good chance of being correct. I think that current Big Bang theories consider that dark matter collapsed before visible matter did
(but do not trust me).
Added later (question evolving): As for the galactic center, even ignoring the fact that there is
usually a very massive black hole there, it is the center of mass.
When two free bodies are satellised around each other (Earth and Moon
for example) there revolve around their common center of mass. But
this does not contradict the fact that a third body can get more
attraction from one than from the other if it is closer to the
former. If big enough, it can also perturb the system.
Indeed the planet Neptune was discovered (actually predicted) by
Leverrier because of the perturbations caused to the
orbit of Uranus.
Rigid bodies can also revolve around their center of mass, in a uniform gravity field, and your barbell might well do that. The galaxy itself can be seen as a body revolving around it center of mass, though this is a bit more complex since it is not a rigid body (I am reaching the limits of my competence).
I guess (literally) a simple way to analyse the galaxy is to
approximate it as a unique body with a given shape (ellipsoid, disk,
...) and a given density distribution. Then you can analyse gravity
within that shape (and outside too if you are a fan of intergalactic travel). This will give you good approximations, as long as
you stay away from large bodies that locally strongly change the
gravitational field, as the end masses of your barbell.
Note also that from far away, compared to the barbell size, it does not make much difference what point in the barbell is considered as center of gravitational attraction.
Last remark : you can actually find asteroïds that have pretty much the shape of your barbell. They do not have a strong gravity field, but computing trajectories is not easy, as it rotates also. It is somewhere on the net, probably Nasa.