Now, if you can come up with a better analogy, fantastic. But the best science communicators in the world haven't done any better, so I'm not holding out a lot of hope for success...
EDIT: I'd also remark that collapsing centuries of debate on what's a scientific theory into "everything is an analogy" is a disservice to the public: good luck distinguishing realism, structural realism, positivism, etc., if you put on the same level science popularizations and proper (albeit imperfect and incomplete) scientific theories.
Also, the science communication bit is a red herring. There are plenty of concepts that cannot be explained easily to the public. One might justify just as well going around popularizing Laffer's curve, since actual macroeconomics is impossible to explain on a napkin.
If one flips the rubbersheet, the geodesics are the same, but clearly now the model doesn't match our intuition. Indeed the objects are attracted by the mass because of the Earth's gravitational potential, not because of the rubbersheet curvature. The balls are following the variation in height of the sheet (in a sense a first order variation) not the curvature of it (the second order variation). The model is bending the wrong component (space, instead of time), and with the wrong sign (the geodesics are repelled by the mass).
As pointed out by Ron Maimon in this answer: http://physics.stackexchange.com/questions/1019/common-false...
a much better explanation can be offered introducing proper time and the effect of gravity on it.
Could you explain it to me this way?
Finally, an explanation of gravity in layman's terms.
Still, there is the kernel of the correct explanation which could probably be presented to the larger public, starting with the well known explanation for refraction in terms of minimal paths (the lifeguard trying to save a drowning man), passing to the fact that in special relativity the trajectory of the free particle maximizes the proper time, and ending up with the gravitational case.
It's not hard to summarize the basic tenants of GR, here is a very good attempt: http://physics.stackexchange.com/a/3018/852. For a layperson I would focus on (1), what a geodesic means (what's the shortest way to travel from NYC to London, etc.). Then I would move to (2) and say that we don't know why that is the case. This would be a nice segue into a discussion about how Physics just describes observed phenomena with equations and the explanations are iterative.
What? It's literally the core of the conversation.
Your argument seems to be "it's impossible to accurately explain GR to people in a way they'll understand, so we shouldn't even try, and any attempt to use analogies that may lead to incorrect intuition should be abandoned".
I fundamentally disagree with that premise. It's the height of academic elitism, and presumes that perfect fidelity that few understand is more important than imperfect analogy that might allow laymen to gain some minimal insight.
The Laffer curve is just wrong on its face.
1. Prices move according to the law of supply and demand.
2. Right: why do they move though?
1. Simple: every hour the central government determines the current supply and demand for all goods, and computes the right price.
It looks like a market economy, the prices move in the right direction (the balls are "attracted" to each other), but the explanation is a planned economy (the effect of Earth gravity).