Perhaps it helps to imagine someone had a "screwy thumb" and the coin only precesses when they "flip" it (in fact people can train themselves to do this, and its very difficult for you, the sucker, to see in the air that the coin is not rotating but just precessing!). Hopefully its obvious that whatever side is initially facing up will be the same one facing up when its caught?

The next step is not at all intuitive to me, namely that even someone trying to do a fair flip causes some precession, and that this isn't decoupled from the rotation.