I'm not sure why people want this term to stick to the Ancients as if that's what it is.
It's literally showcasing that the Ancients could be physically immortal if they wanted or needed to be, but that they still don't consider themselves above what's supposed to be the natural cycle of their planet. To them, their souls are its blood. Without blood, the planet dies, and them with it. To return to the star is to ensure that the star stays alive for future generations.
It's the softest form of "death" imaginable, because they got to choose when it happened. Where it happened. How it happened. Who got to witness it. Maybe throw a party for it. With the full knowledge that they may very well stick new feet into the mud again one day, with a new face, and a strange affinity for the same favorite foods.
I basically see them as being akin to the Valar. Convocation of 14 and all that. There's just no Eru in this setting. And the timeless halls are within the planet, rather than without.