You are stating the analogy just fine, but the limits of analogies come into play here because the sheet has no propagation speed and it has no movement of its own, either (nor does the big ball at rest in its center).
Your sheet even has bounds, something that we aren't trying (on this thread) to determine for Gravity.
What we want to know is how fast Gravity's influences propagate. If the Sun simply disappeared, would the Earth shoot off tangentally immediately, or would there be an 8.3 minute delay?
You are stating the analogy just fine, but the limits of analogies come into play here because the sheet has no propagation speed and it has no movement of its own, either (nor does the big ball at rest in its center).Those are the aspects of the situation that are analogous to the sun.
The sun, insofar as it is not subjected to accelerations, is at a fixed location in its own reference frame. The sun's gravitational field, as long as the sun hasn't undergone an acceleration for more than 8.3 minutes, is fixed to all points in space, at least as far as the Earth's orbit is concerned. The field, like the sheet, does not need to propagate; it's already where it needs to be. The Earth will "see" the same field at every point, regardless of how fast it's moving (neglecting the "gravitomagnetic" effect, which is a known relativistic correction that is vanishingly small).
Should the sun undergo an acceleration (or, God forbid, disappear), the changes in the field will propagate from the sun outward. These changes are gravitational waves, and they propagate at c. Again: waves propagate, fields do not.
The situation is exactly analogous to the electromagnetic field vs. electromagnetic waves. The same geometrical argument applies. Van Flandern now says that while electromagnetic waves propagate at c, electromagnetic fields propagate infinitely fast. Whatever. It's like saying that while cars drive at finite speeds, the road goes infinitely fast. I suppose that's one way to describe it...just not a very useful way.
(I had promised myself not to respond to this Van Flandern crackpottery any more, but RadioAstronomer has inspired me. For the future, I'll put together a boilerplate response.)
Yet another picture, I may go blind.