The clone stamp would copy another part of the frame - it might look similar, but unless it's just a plain background with very even lighting, it is probably not the same. In one frame it might look ok, but when you play the sequence, there will be a ghost of something moving where the rig was.
I got this effect when erasing rigs on 16mm film footage, and I've seen it in other productions. (I used to call it the Predator Effect, a sort of almost-invisible-but-not-quite, like the cloaking device in the film. You didn't see the alien, you saw the jungle behind it, but with a little distortion.) Because with 16mm film transferred to video, there was a slight movement of the frame, and the film grain was different each frame, erasing through to a clean background plate or to another frame had a very subtle difference. I made a Sugar Glider ( a possum something like a flying squirrel) leap off a tree branch into the air, and had 3 fishing lines holding it up. The line caught the light and showed up. So I only had 3 very thin lines to erase, but you could see them as they moved across screen. On any one frame they were invisible, but the motion revealed it.
Frames shot with a DSLR are steady, there is no movement, and no grain (or noise) if you use enough light or long exposure, so the exact same part of the frame should match, and you can erase the rig to see it. But a different part of the frame probably won't match. You would get away with it for one frame, but with a whole sequence where it moves across the screen, I don't think so.
Maybe you can source the area to be cloned from one layer, then stamp it on the other layer in the same place (you can in Photoshop), but you would have to be accurate, and why would you bother?