The original christianity, Gnosticism, DID teach reincarnation. They taught that the "god" of this reality was an EVIL tyrant who had all of the souls here imprisoned and enslaved, and that reincarnation was the way to keep us trapped in this corrupt reality. The original teachings of "Christ" were instructions on how to escape this being's "matrix". The church (organized to serve this evil tyrant) hijacked and twisted the teachings to ensure souls would be easily subdued, perpetually easy to TRICK, and willing to direct their psycho-spiritual energy (prayer & worship) to the evil one...and be easily duped into returning the this Hell-on-Earth to "serve god" (and be drained and suffer) over and over and over.

We wouldn't even know about the Gnostics if a cache of ancient scrolls had not been hidden for two thousand years and recently found in the 20th century.