Firing Hans seems dangerous. They wouldn't want somebody feeling jilted and getting upset at them and going to the FBI. If he was a really bad spy (like Paige), there's a lot they could give him to do. Same for Paige. Since Elizabeth sees her as a "9 to 5" spy, there's no real reason to drag her to all these operations. I think Elizabeth is trying to create a fantasy of her daughter going in her footsteps.