Again, this is kind of the issue with people conflating kegare with death. The presence of kegare is what influences "change" (in the abstract) and therefore deterioration, sickness, death, etc, and being "around" that kegare spreads it to you. But it doesn't follow that removing life and death from yourself should make you free of kegare, that's backwards. I also don't think "adding the concept of death" quite gets the point across, even if it's kind of accurate? It feels to me more like they're thinking of some TYPE-MOON-like thing where now the thing is related to death in some abstract capacity.
It's mused upon that kegare spread throughout the Earth by valuing life over all, and killing others in order to survive. Things on the Earth treasure life over all, and by doing so taint themselves with the idea of death; if living is the best, death is the worst. If you need to kill to survive, you spread the death of others by valuing your own life, spreading that taint throughout the world. But it's suggested that if only life wasn't treasured as such, the taint of deterioration and death would not have taken hold, and lifespans could be however long.
This leads into both of the concepts behind the Hourai Elixir and the Lunarians. The Lunarians figured that kegare, this taint, was what influenced the deterioration and death of the world. So just as beings from the ocean came up to the kegare-free land, the Lunarians moved to the kegare-free Moon. The difference is that they stopped valuing life itself and killing others to maintain it, so the taint of kegare doesn't spread. Meanwhile, taking the Hourai Elixir is giving into the temptation of eternal life, valuing life above all and rejecting death, which gives you that very same taint. This is partly why the Lunarians have been said to test humans with the Elixir, to see if they value life itself over living. (EDIT: Following was just a side comment, see Clarste's argument) Eirin says that a human taking the Elixir invites an eternity of suffering, where you are not even allowed to die, unable to live with humans as a human. It's seen as an awful existence in general, which I suppose is why Eirin feels so ashamed about Kaguya's fate.