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.
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.
"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
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 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.