The only time you should insert anything into dialog in this manner is when you are explaining something about how they are speaking.
Good: “The unified realm of my childhood,” she explained tensely as she paced in front of the audience, “had one councilor for all the districts.”
This gives information about the manner she is speaking and so guides your reader on how to understand the words. Maybe you could throw in some gestures like stroking her chin or clenching her fists or even indicating she had a dry mouth and so licked her lips to moisten them. She could yell or whisper or speak hesitantly or like a priest chanting a religious service. There are many other things.
Don't get too excessive about it. Keep in mind that your reader has to follow you through the text to get the meaning of the sentence. Dialog should flow like a real person speaking. So you can insert things like indicating the speaker has paused and is doing something during the pause.
Good: “The unified realm of my childhood,” she explained while serenely stirring her tea, “had one councilor for all the districts.”
Again, this paints the picture of how the speaker is behaving during the dialog.
There is not an absolute barrier between good and bad. Generally, the farther from describing the means of speaking, the more potentially troubling the interjection.
Borderline: “The unified realm of my childhood,” she mused while recalling the festivals and fields of her youth, “had one councilor for all the districts.”
This one is potentially useful since it might indicate the speaker had a dreamy or distracted attitude. It would make sense if the speaker had an established history of speaking this way. Or you were working on establishing such.
Your example goes too far afield.
The unified realm of my childhood,” Lucy hoped to avoid discussing the reasons for its disintegration; the children carried the blood of the traitor, and she carried more than her share of the scars, “had one councilor for all the districts.”
You are talking about her hopes rather than her manner of speaking. And you are talking about other people and their characteristics, people that are not speaking.
If you want to get this information in, you should expand the dialog, or put the description outside the dialog. For example, if she wants to avoid a subject, show her avoiding the subject, possibly have somebody call her on it. Or have her explicitly say she does not want to talk about a subject and have her give a reason.
Or have somebody else explain that the subject should be avoided.
The method you choose should be one that advances your story.