To explain why Asura and Indra are important: The Rinnegan is not a natural power. It is a power gained by the Ōtsutsuki clan when they consume the Fruit of a God Tree and become Divine Beings. Except in Kaguya's case, becuase she ate the whole thing instead of bringing it back to their family to be made into the cinnabar elixir(incidentally, elixirs made of cinnabar were another method by which Taoists pursued immortality,) so she gets the much stronger Rinne Sharingan.

It's basically a mark of Divinity, you become a God by consuming the God Tree's fruit: In the databook, the Sage of the Six's paths eyes are listed as a Kekkei Mora rather than a Kekkei Genkai.

The Sage of the Six Paths inherited this power in a diluted form from his mother, giving him his Rinnegan, and Ashura and Indra each inherited half of this power from him. Indra the eyes and Yin and Ashura the chakra and Yang.

The divinity, it seems, is in their chakra and is diluted too heavily in their Descendents in later generations. Recombining their bloodlines allows for the Sharingan to be evolved into the Rinnegan, but this is only the physical eye, not the divinity, so it's weaker: Madara was not able to cast the Infinite Tsukuyomi or even use the most advanced powers of his Rinnegan with the fake eyes that Kabuto equipped him with.

Recombining their chakra will replicate the "Six Paths Power," which is to say, their father's divinity, which combined with a fully powered Sharingan will recreate his specific Rinnegan. It should be noted that the other Ōtsutsuki have Rinnegan that do different things. Momoshiki's let him absorb techniques and send them back strong as well as turn people into pills that he could eat to rejuvenate himself and gain their powers. Momoshiki's lets him warp space, reverse time for a few seconds, and gives him a flamboyant version of the substitution Jutsu.

When I say that Senko's would be an "artificial one" that's weaker than the real one, based on canon, I mean that she wouldn't be able to do all of the extreme bull-crap from late in the series that we're looking to avoid. No summoning meteors or the like. If necessary, I can say that the imperfect, ad-hock mixture is tainted by the traits of other bloodlines she grafted herself with so her Rinnegan has completely different powers and can't access the Six PAths at all. (In which case it will probably be amber instead of lavender.)

If you want to reverse the non-reincarnations thing, that's fine, but I was just saying that it means that we can use the Rinnegan without getting too overpowered becuase it's canon that not having the Sage's power of the combined power of his sons canonically makes it weaker.

(The irony being that, if you do reverse the reincarnations and make Kimiko the inheritor of Indra's chakra, that would mean that at some point her Uchiha hating clan would have had to have interbred with the Uchihas.)