Let's see if this helps at all. I have digital copies of the complete manga
series (so everything here only has to do with the manga and no other version), and I went through the last and first chapters of Stars to find this. Beware, everything in this post is very SPOILER-HEAVY for people who have not yet read Stars.
In the very last chapter of Sailor Moon Stars, Sailor Moon sacrifices herself into the Cauldron in an attempt to trap and overwhelm Chaos there. All of her comrades have already died and had their Sailor Crystals thrown into the Cauldron; Mamoru is shoved into the Cauldron, and Sailor Chibi Moon fades completely out of existence. Even Sailor Galaxia no longer exists. Only Sailor Chibi-Chibi (who transforms into Sailor Cosmos) and the Sailor Quartet remain throughout these scenes. Sailor Cosmos explains to the Quartet Sailor Moon's intentions with her final sacrifice, her (Cosmos's) own history of running away from her home when her battle became too overwhelming to her, and she explains some of the nature of the Galaxy Cauldron and the stars/ Star Seeds. (All quotes following come from the Miss Dream scanlations.)
The way Sailor Cosmos explains it, it would seem that the power of rebirth comes from the Lambda power through the combined power of all Sailor Crystals together. Additionally I believe it is important to note that the terms for "star" (sometimes different words/ pronunciations, but the same kanji) used in Japanese is often used more loosely than in English; I've seen it used to refer to planets as well (as evidenced by the Japanese names of planets, such as Mizusei
-- "water star"-- for the name of planet Mercury). I believe I've read somewhere that it can be used to refer to other objects in space, such as meteors, asteroids, etc.-- hence, the Asteroid Senshi/ Sailor Quartet.
It is well-established in the Sailor Moon universe that all Sailor Soldiers are the embodiment of the power of their corresponding planets/ moons/ cosmic bodies. This is why canonically there are no Sailor Soldiers of gemstones, flowers, rainbows, etc. as is popular in fan fiction. Judging by context from reading throughout the manga series (this is my interpretation of it), it seems that for every cosmological body in space, there is a Star Seed which is created with it, and this Star Seed is/ becomes/ evolves into/ combines with a Sailor Crystal, and its wielder is that body's protector, from whence she draws its power. Thus, the Galaxy Cauldron first creates a Star Seed, ejects it into outer space where it transforms into a star/ planet/ asteroid/ comet/ moon/ whatever (over what length of time, I couldn't guess), and someone is born/ created on that planet who holds that particular Star Seed, which becomes a Sailor Crystal for that person to draw her power from. From here, I'm guessing that when a Star Seed is destroyed (its owner/ weilder is killed), the Star Seed is automatically drawn back into the Cauldron for recycling, sent back out again to restart/ continue its existence in order to protect that planet and continue the cycle of life. (I suppose the alternative would be for that entire planet to completely cease to exist in the prolonged absence of its Star Seed, but I'm not sure.)
Looking further back in Stars, I found this quote from Sailor Iron Mouse upon her first appearance:
Iron Mouse: "The Last of the Star Seeds, the holders of the Sailor Crystals... I can't believe it, the other ten of them are here, too!"
"The last of the Star Seeds on Earth! The Sailor Crystals!"
This implies that ONLY a Sailor Senshi can ever hold a Star Seed; no other beings can possess one. This furthers the idea that only those with a Star Seed/ Sailor Crystal have the capacity to be reborn, because they are the designated protectors of their respective star/ planet/ body.