There's two parts to the situation you describe I think. The first part is the therapist's stuff being present in the room and the second part is the client taking responsibility for managing that stuff for them. Whether that be trying to avoid triggering them or whatever else.

My T's stuff sometimes comes up (as you know) but I don't take responsibility for managing that for him. In fact he very explicitly says he doesn't want me to take care of it for him. That's the source of my impatience with him at the moment- I feel he needs to manage it better himself, because it's not my responsibility to censor myself for his benefit.

To answer your question, it's not necessary to take care of your T, and for me, it would be a sign that something was seriously wrong if the therapist were to encourage or accept the client's efforts to manage the therapist's material. The therapist should be managing that themselves in supervision.

So the desire to manage T's stuff is normal in my view, it's the therapist's actions that can be problematic.