Rav Elyashiv retired from the Rabbanut, as mentioned. He had only one interest after that: to learn Torah. As R Schachter explains, when you are a Dayan for the State, your context is as wide as your constituency. You will necessarily consider certain leniencies in keeping with the audience of your opinion. The decisions back then are published by Heichal Shlomo.
Once he retired, he didn’t put out an advertisement which said “Psaks-R-us” or seek a new appointment. Rather, because of his reputable erudition, people gravitated to him. His home became a magnet over time for major questions, and eventually the not so major. His audience was now very different and the strictness in his approach consistent with that.
In that context it made absolutely no sense to appoint a “successor”
He knew there were other Poskim that people could and did turn to. He chose not to impose a successor.
I would argue that this might have been one of the most profound Psak/legacy that he left us!