I was going to write an incorrect answer at first, but having researched a bit
what Yukterez wrote, I think I may be able to offer some intuition.
First, let's look at the solar system in the centre-of-mass frame, where the
Sun is essentially stationary in the middle. Here, the force on the Earth is
directed towards the Sun's position 8 minutes ago, which conveniently is the
same as its current position. So this doesn't affect the Earth's position in
the way you say.
If the Sun and Earth were moving at a constant velocity to each other, they
would, like Yukterez wrote, still be attracted to the instantaneous position of
the other body. This is not because of a faster-than-light influence, but
because gravity is more than a simple attractive force between objects
(analogous to the electric field in electromagnetism). For moving sources,
there are other components of the gravitational field (roughly analogous to the
magnetic field). You can view this as the force travelling at the speed of
light, but being directed towards the position of the source "predicted" based
on its past linear motion.
However, if the source is accelerating, like the Earth around the Sun, the
effects are less obvious. Then, you really can't view it only as an attractive
force towards any position (I think, correct me if I'm wrong), but you get
more complex things like gravitational waves. But still, nothing ever travels faster than
As I am more familiar with electromagnetism, let's take an example from there.
If two oppositely charged bodies ($A$ and $B$) are moving with constant
velocity towards each other, we can view the system in the rest frame of $A$.
Here, it emits only a static electric field, given by the Coulomb formula, and
$B$ therefore feels an attraction directly towards it. In the rest frame of $B$,
this means that the attraction is towards the instantaneous position. But how
can this be since the electric field is propagating at a finite speed? The
answer is that since now $A$ is moving, it also emits a magnetic field. While
this doesn't affect $B$ directly (since $B$ is stationary here), it does affect
the electric field and makes it point in a different direction (towards the