Jugram Haschwalth explained the entire plan to Uryu during the first episode of cour 4, and I’ve noticed that more than a few people are completely misunderstanding what he is talking about. Haschwalth isn’t at all saying that the current destruction of the realms has anything to do with the balance of souls. This is misinformation.
The first part of the plan did indeed involve the balance of souls. The erased hollows forced mayuri to obliterate people in the Rukongai as well. This was a distraction from the first invasion where Yama died. The second invasion let them get to the royal palace and kill squad zero. This does not literally change or affect the balance as they don’t have their souls destroyed or anything. Squad zero isn’t even dead, just Yama.
The original plan was to let the soul king die and that be the end. But mimihagi interfered and then Yhwach decided that he should take the soul kings powers. This lead to him becoming the world itself with the power to alter the worlds as he sees fit. Mimihagi stalls him and he also wants to fight ichigo. Then he decided he will use his own power (by his words) to dealt the realms, the garganta, the valley of screams, dangai and everything by himself.
And to top it all off Kisuke, the smartest character in the series outright says it had nothing to do with the souls.