What's the point with that quite lackluster story in Endwalker?

[Absimiliard]

Nope it marks them as impartial and trustable advisors.

The white robes only mark her as an advisor. Hythlodaeus took them to mean she would be trustworthy and impartial. While a certain degree of assumption of impartiality and trustworthiness can be made based on the advisory role she retained (as the Convocation probably wouldn't have awarded it otherwise) after departing the seat of Azem, those are none the less not the meaning behind the robe.

Doubt the ancients' society would've been down with going extinct anyway, judging by how hard they fought to survive the Final Days. As for Venat, while her advisory position does mean others are more likely to listen to her, events in Elpis rather clearly demonstrate that even she needs to be able to back up her assertions with fact if she expects the Convocation to take her seriously. Kinda hard for her to do that without spilling all of the proverbial beans.