The Mattress, Bowling Ball, Marble Illustration

So many times I’ve heard this illustration of Einstein’s general theory of relativity: gravity occurs because space itself curves around massive objects. It’s like if you put a bowling ball on a mattress, and then roll a marble near it--the marble will be sucked into “orbit” around the bowling ball, drawing closer until they collide.

For the longest time, this illustration bothered me because aside from the curvature of the mattress, what pulls the marble down toward the bowling ball is gravity. How can you use gravity to explain how gravity works?

But as I thought about it more carefully, I guess it’s not really a major flaw. The problem is, the illustration is trying to explain something that happens in “the real world,” which we generally think of as being in three “spatial” dimensions (plus time); but the illustration envisions three-dimensional objects (bowling ball, marble) on a two-dimensional plain (the surface of the mattress) which has been distorted into three dimensions (that surface is no longer flat, but now has a roughly conical indentation). I suppose the role of gravity in this illustration is really just to keep the marble on the curved two-dimensional plain. (Because without gravity, the marble would continue floating forward in a straight line and leave the plain.

I guess the real problem is that it’s so difficult to visualize the distortion/curvature of a three dimensional field (because that would mean visualizing in four dimensions), so we just have to envision a two-dimensional plain being curved into three dimensions and use it as an analog, but then for the sake of making it easier to visualize, the illustration involves three-dimensional objects, which is a little confusing if you think too much about it, as I have.

But the other thing that always bothered me about this illustration is that it assumes the bodies are in motion relative to each other. Isn't the force of gravity just as powerful between two objects that are stationary in relation to each other as between two objects in relative motion? Maybe some smart person who understands such things can explain.

Further remarks added now after some comments have been made: 


Okay, it seems I did not make the original question clear enough. Let me try again.

The illustration assumes that the two bodies are already in motion relative to each other. The only motion that takes place is not due to the "force" of gravity, but due to the marble already having been set in motion by some other force. Gravity in this illustration does not cause any kind of motion--all it does is to redirect the path of something already in motion. There is a reason why persons offering this illustration will always say that the marble was rolled near the bowling ball, rather than saying it was placed on the curved part of the mattress (because in the latter scenario, they really would be using gravity to explain gravity). If the analogy worked properly, placing the marble on the curved part of the mattress would have no effect; they would both remain stationary in relation to each other. The only reason there is any motion in this illustration is because some unknown force set the marble going to begin with. Gravity does not "pull" the marble, and does not cause motion of any kind--all it does is affect the direction of something that is already in motion.

My question is, in real life (not the analogy), isn't gravity able to cause motion? If you were to place two objects near each other (not moving) and if they were far enough away from other massive bodies that the force of gravity would be greater between these two objects than between them and anything else around, would they not start to move toward each other? Newtonian physics seems to say that they would. I don't know about actual general relativity, but the illustration used to explain it implies they would not.