The points are actually on the very top of each wing, it's extremely finicky. kinda like giving each wing a headpat if that helps lol.
Actually the free hand is so buggy and terrible that the game is probably better off getting rid of it in place of a simple dot cursor instead. and there's no need for any free control in the Deep Massages, since its the same three points every time and you can't interact with anythign else. It may as well just be you selecting between the three clearly visible points, no need for the free hand.