Personally, I don't see proving reincarnation, nor finding flaws as needing philosophical discussion.
Reincarnation, as a historical belief, I see as no different than believing in resurrection or an eternity spent in the Kingdom of Hades. The only difference is that people are not, to my knowledge, claiming to have been resurrected on a regular basis nor coming back from the brink of death with accounts of personal encounters with Hades. By taking a simple glance at the current cultural belief I can see it as nothing more than a modern trend.
I have personally gone from being a full believer in reincarnation to a doubting Thomas to quite shaky. (I'm probably repeating myself from some earlier post I can't remember right off the bat.) In all my experience I have not encountered a single, testable definition of the soul, let alone reincarnation. Nobody seems to agree what it is, unless they are a part of organized religion. Then the soul has an agreed upon form. But that's not science. That ends that. No testable hypothesis, no science. Yet, even without a clear definition of the soul people still try to gather evidence.
Proving reincarnation, in our culture, has tended to revolve foremost around providing memories. The trouble with claiming memories is that one needs to find verification that such memories actually took place. The most obvious conclusion that I've encountered in believing circles has been to read a biography or other historical resource. But the confabulation of memory and cryptoamnesia are two verifiable phenomena that have not, to my knowledge, been satisfactorily dealt with in proving that claimed and verified memories are indeed evidences of reincarnation. From the start we have no idea the process that someone came to believe their claim. The only controlled, even semi-controlled, setting that I know of in attempting to provide evidence of reincarnation has been hypnosis. I don't know hypnosis to be a reliable source for good scientific experimentation.
"At this point it is impossible, without other corroborative evidence, to distinguish a true memory from a false one."
This was quoted from the American Psychological Association. (I got this from the False Memory Syndrome Foundation website. I don't know if this counts as copy-righted material in its entirety.) The issue is quite simple. If the experts can't, so far as this quote indicates, distinguish true and false memories without corroborative evidence, how can non-educated laypeople, who seem to make the bulk of adherents, who regularly set up websites, forums, and write books?
Barring the insane, this elephant in the middle of the living room has been tackled by various believers that I've known personally. They find others who share their experiences. This has been, with some, the best form of measurement of the reality of a past life. (I won't go into details because their info is private.) The one major dilemma in attempting this situation is that for a past life that occurred far enough in the past that direct observers are currently deceased, you are very likely relating to another reincarnation claimant with the same issues of confabulation and cryptoamnesia that could easily invalidate their claims.
I'll end by saying this. The most grounded, most verified, most intelligent, most articulate, and most reasonable claimants that I've known (including myself) have all had either some background in paranormal beliefs or extracurricular paranormal beliefs that underscore our reincarnation beliefs.
I think that sizes it up.