Very Shaky camera but a constant background
While it's fairly easy for it to figure out the camera position from the shot, it is not nearly as easy for it to identify subjects cleanly.  In order to do what you are talking about, the subject would have to be identified and extracted from the background without any extra bits coming along.

Computer vision algorithms simply aren't that advanced yet and such work is always either done by rotoscoping (manually outlining the subject on each frame) or using a chroma or luma key backdrop that can be removed.  Stabilization works by identifying straight lines and tracking them generally.  The border fill in motion warp then works by attempting something similar to what you are talking about by looking at what appeared to be there in previous frames, but it doesn't really work all that well yet either as it is a new technology.

The same problems that prevent them figuring out accurately what was there in previous frames are what prevent separating the subject from the background (the ability to identify, with precise accuracy, the same object in one image to another.  You would end up with a subject that hovered oddly on the background and had a ring of background around them and/or left bits of them behind if you tried it with current level of technology.

Using motion warp and applying a crop or bypassing the auto-crop and manually cropping and moving around the frame is probably your best bet.  This is how I usually handle stabilization issues.