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.