Balls on sheet is a wrong model imo. It needs gravity to work and therefore doesn't explain gravity.
I like to imagine a sponge. If you could somehow make dense lumps inside the sponge (may be apply heat in its center somewhere using microwaves?) everything around that lump will be feel a tension/attraction towards that lump. That's my mental model.
I think it’s a good model because it poses the same question: an object is in a field gradient of otherwise forceless space, so what? In 0g a mass on a rubber sheet won’t attract an object, so why would real gravity “lines” attract an object that doesn’t try to move across it? There’s no supergravity for “gravity sheet” to work either.
Afaiu, the answer is that moving through time in a gravitational gradient makes you constantly “fall flat” on an x/time 2d graph due to time dilation gradient, thus affecting x and spatial momentum. Apparently mass affects time and time creates gravity. Idk if that’s a fringe idea or mainstream.
in 0g, object that does not leave contact with the curved (by-whatever-means) rubber sheet will behave the same way
"using gravity" in the experiment is convenience of the experiment - gravity is freely abundant to use, both as a source of curvature and source of "making objects not leave the sheet"
if you want to hold the sheet by yourself, and hold the objects to the sheet by yourself - you are free to do it, enjoying your victory of "not using gravity"
but the point is not in the means, but in the end - create curvature, keep contact
in 0g, object that does not leave contact with the curved (by-whatever-means) rubber sheet will behave the same way
An inert object will sit on it kissing it and never go anywhere. The curvature of a sheet doesn’t matter cause there’s no “down” and no forces. If you push it “in” a little and release, the sheet tension will push it back out but not “to” the well. You may start accelerating the system to create gravity-like situation, but that’s the point.
I think the balls on sheet is an OK compromise. The reason for me is, it's tough (but not impossible as you point out) to visualize functions over 3D space. Your sponge does that and I like it! (You could also use color-coding, vector-gradients etc) The ball and sheet uses a spacial dimension as the function value, and that's the dimension the needs gravity to work acts on. So, if you accept that as a compromise, it's OK; we are saying that dimension is a convenience.
That helps explain curvature of space, but it doesn't explain any of the motion.
When you see those stretched rubber sheets, they rarely/never show the grid lines moving. So let's say you place a small *stationary* object at a specific grid point (coordinate), and there's a large object that deforms the grid itself. Nothing about that visual provides any new intuition about why the small object would leave it's current grid position and move towards the larger one.
The visual tries to explain gravity by appealing to your existing notion of gravity.
Sure, but the problem is that the geometry of the sheet alone doesn't indicate what will happen to the matter on it; you need something external. Whereas in actuality the curvature of spacetime alone is sufficient.
I like to imagine a sponge. If you could somehow make dense lumps inside the sponge (may be apply heat in its center somewhere using microwaves?) everything around that lump will be feel a tension/attraction towards that lump. That's my mental model.