Inherited traits are passed on in our DNA, such as hair color, eye color, etc. But each person is unique unto him/herself and can never be duplicated exactly which would have to be the case for being reborn in another body as a copy of a person, even mentally and emotionally. In fact this is the function of evolution. The joining of different DNA insures variety and avoids exact duplication. That is a good thing. There are species which clone themselves into identical copies, but none of them are human and even as they are physically the same, their behavior varies from individual to individual. The silvery salamander is an example. But then this salamander does not use male sperm and each offspring is female and an exact clone of it's mother. As to a spiritual rebirth in a different body, that is already happening, family traditions, environmental demands, societal cultures are all contributing to a duplication of behaviors, but those are always learned, and not inherent at birth or shortly thereafter. Our "persona" is formed by every piece of information we absorb from birth on, this is how the spiritual (mental) information is transferred. It is a function of out "mirror neural network" in the brain. But it starts pretty well as a blank slate except for keeping the body alive (survival). But their is no empty mind looking for a soul, or a soul looking for an empty mind. That is a religious assumption, such as possession. Besides the argument in the OP is self defeating. Humanity is growing by an exponential function, but if there is a limited number of souls, how does the number of souls increase? It is obvious that there would always be a shortage of souls and then what? Zombies? It is clear that if there are souls, each person has a different soul and new souls are being created all the time. To think there are "super souls" which can transfer into a mind is hubris and speaks of a lack of understanding of the way things work.