On the surface, you do have a lot of emotional power over the players at your table. Depending on the turn of a card, they can have their hopes crushed, or their outlook on life can soar. But at the heart of it… it’s the cards creating that magic, not you. You are merely the messenger. And like most messengers, you will often get shot (hopefully only figuratively) when things don’t go the player’s way.
Otherwise, you have zippo in terms of power. You are told what tables to work, when to work them, and for how long. The way in which you must deal each game is exacting and demanding, and there are cameras everywhere you look that are recording your every action and making sure your performance is up to snuff. (Reminds you of that time you babysat for those uber-paranoid parents, doesn’t it?)