I'm motivated to repeat something I posted in some other thread but tailored to this one. While it's true that using the tools of science we know about today, we can't prove mechanism, in these cases we can falsify at least part of the the claim of reincarnation because all of these cases can be checked carefully and the claim that some young children have memories of the traumatic death of someone else, but also have birth marks/defects corresponding to that death and that there is no way to explain this with current scientific knowledge.
And the second point is that, as with string theory up to recently, there's no way of falsifying the theory currently but we should not dismiss it out of hand for that reason.
So to me a reasonable scientific approach would be: there is some evidence for which survival of some part of us after death is an explanation. But we don't currently have a way of proving or disproving this.
Ian Stevenson’s Case for the Afterlife: Are We ‘Skeptics’ Really Just Cynics?
Stevenson’s magnum opus, published in 1997, was a 2,268-page, two-volume work called Reincarnation and Biology. Many of his subjects had unusual birthmarks and birth defects, such as finger deformities, underdeveloped ears, or being born without a lower leg. There were scar-like, hypopigmented birthmarks and port-wine stains, and some awfully strange-looking moles in areas where you almost never find moles, like on the soles of the feet. Reincarnation and Biology contained 225 case reports of children who remembered previous lives and who also had physical anomalies that matched those previous lives, details that could in some cases be confirmed by the dead person’s autopsy record and photos.
...
First, he was convinced that there is only a brief window of time—between the ages of about two and five—in which some children retain these reminiscences of an earlier self. Importantly, their statements are, in principle at least, empirically falsifiable.
Then as a second point, there are arguments that falsifyability
can be taken too far: “
I think falsifiability is not a perfect criterion, but it’s much less pernicious than what’s being served up by the ‘post-empirical’ faction,” says Frank Wilczek, a physicist at MIT. “Falsifiability is too impatient, in some sense,” putting immediate demands on theories that are not yet mature enough to meet them. “It’s an important discipline, but if it is applied too rigorously and too early, it can be stifling.”