No, it sounds like you want to do it because you feel like the benefits of therapy should be mutual and you don't feel like you're pulling your weight. But that's not how therapy is supposed to work.

This is why people are suggesting there is more to explore here. There is no reason for you to feel awful. I think you just find it uncomfortable for the caregiving to all be focused on you. Think about going to see a doctor. You don't compensate them by spending time discussing their health and wellness. When you go see an accountant, you don't spend time talking about their finances or comforting them if they are struggling financially. And you definitely should not spend time comforting your therapist over things that happened to YOU because SHE gets upset about them.

I think maybe you are confused about the nature of the relationship due to your therapist's crying. Professional relationships are not about providing mutual support to each other, except in the sense that clients provide financial compensation in exchange for a service.