That's exactly what I was thinking, but make it more like a "spinning top" bottom whereas the point of the cone is sharp enough that you don't want to use the palm of your hand to insert it into the bottle.
Also, make the lip of the bottle stopper smaller in diameter than the diameter of the bottle opening so that the user cannot grab it around the edge.
This is an excercise in designing a pretty object, but that is totally useless.
BTW I'm using a simple silicone stopper end and not a nice stainless steel stopper.