From a simple google search:

• Reincarnation is the religious and philosophical belief that the non-material essence or soul of a living being begins a new life in a different physical form after death [1].
• The Cycle of Rebirth: Known as samsara in several traditions, life is viewed as an ongoing cycle of birth, death, and rebirth. Karma: In many traditions, actions in one life directly shape the conditions and form of the next existence through a universal law of cause and effect[4].
• Liberation: The ultimate goal across many faiths is to break free from this continuous cycle — referred to as moksha or nirvana — rather than to keep being reborn [3].
• Hinduism: Teaches that an eternal soul (atman) transmigrates through various bodies until it achieves liberation and unity with ultimate reality[5].
• Buddhism: Rejects the idea of a permanent individual soul, instead teaching that accumulated karma and consciousness (vijnana) pass on to a new existence [2].
• Jainism and Sikhism: Both embrace doctrines of karma and successive existences tied to spiritual purification.
• Western Religions: Major traditions like Islam and Judaism generally reject reincarnation, focusing instead on bodily resurrection or an afterlife with final judgment.
• Incarnation means taking on a human body or becoming "in the flesh". [6]
• Atma (or ātman) means the true, inner self, eternal spirit, or soul in Indian philosophy. [7]
• Reincarnation generally involves a complete loss of conscious past-life memory [8], though certain traditions and case studies suggest fragments can occasionally surface [9].
• Evolution is the change in inherited traits of living things over many generations. [10]

What if reincarnation was supposed to describe the impact of one generation's karma on the next?

Let's consider an alternate hypothesis: That every living being gets re-incarnated as their offspring.

Children would then be considered the incarnations of their parents. Or explained another way, parents can be considered to be reincarnated as their children [1, 6].

This does of course mean that there isn't a 1-to-1 mapping between the life from a cycle and the life in the next cycle; which is traditionally/religiously assumed, but not stated.
To clarify: parents (2 different lives from a cycle) may share many children (incarnations in the next life).

Parents memories do not pass on to their children [8].
Other non-material essence shows itself in the likeness of children to their parents; passed from parents to children via DNA [1].
It also not difficult to believe that accumulated karma and consciousness pass on from parents to children; and from a generation to the next [4,2].
The theory of evolution describes how key memories and consciousness are passed on via DNA [9, 10].

Perhaps the cycles did not have a clean start (birth) and end (death)[1].
This discreet start and end could have been a simplification brought about in the story telling to make it easy to explain.

Any person who does not have children, ends a rebirth cycle [3,5,7].
This could explain the origins of religiuos celibacy.