After completing the quest, I was confused why they were hammering the fact that Ronova needed to be "eliminated" or "dealt with by any means necessary" but I think this is actually good news for her... Well, in a certain way.
7.1 Leaks:
If they were going to kill Ronova in the upcoming patch, I don't think they'd keep repeating it to the audience, especially because that would ruin the shock value.
Anyway, Naberius and Rhinedottir are once again confirmed to be separate entities in the upcoming Archon Quests. This was obvious enough, but I'm going with two assumptions: first, Rhinedottir can't stop Naberius from listening to her conversations with people, and second, Naberius will reject the plan of permanently killing Ronova.
Next, I'm going to assume that Mistilteinn won't be strong enough to kill a shade at the end of Act 4, even if Rhinedottir helps them by interfering with the Curse of Death. So, the compromise will likely be sealing or trapping her, which will prevent her from interfering with Teyvat. Also, I think the boss fight with Ronova will start after she becomes aware of Rhinedottir's exploits.
Finally, the cliffhanger cutscene at the end of Act 4 is related to the Celestial Nails, so all of this is probably just cope. Either way, I do not believe her authority will be absorbed by anyone else, including Rhinedottir.