"Approaching in public" can mean so many different things. What most women have a problem with is being approached in a situation where the man has some form of power over her. For example a cashier or server has to be friendly and smile or she will risk her job. Being approached is uncomfortable, because too many men don't understand it when the no comes with a smile. They are also in a position of power, because they can complain about her or make a scene and get her in trouble.
In this case the TSA agent has some form of power over her and could give her trouble if she refuses. She has no way of knowing if a "no" will be met with understanding or with him holding her up, being insistent, keeping her from passing. That's what makes it especially uncomfortable.