Reincarnation is a priori unlikely given what we understand about the physical basis of the brain.
The evidence is also quite shaky, especially given well-documented instances of confounding phenomena (like implanted memories). As soon as you start asking: "How do you know that the account is accurate? How unusual are these details, anyway--should we even be surprised? How do you know there wasn't any undocumented contact? How do you know the child wasn't exposed to information about the other person without others really being aware?" you find that...well...surely there wouldn't be so many cases like that, right, not with a highly motivated investigator trying to find as many as they can? Surely not so many. They couldn't all have flaws like that.
And the interpretation is overly constrained.
Maybe it's not reincarnation; maybe the universe has attractors for some patterns, and this both causes people's lives to turn out some way and causes people to imagine such things. (Lots of people believe in "luck".)
Maybe it's not reincarnation; maybe it's a type of clairvoyance. (People believe in various sorts of visions.)
Maybe it's not reincarnation; maybe the child is visited by the spirit of the dead person. (Lots of cultures report ghosts and visits by spirits.)
Maybe it's not reincarnation; maybe the children's thoughts have changed reality and the past to make everything else conform with what they think. (How do we know our present thoughts don't change the past, anyway?)
Maybe it's not reincarnation; maybe it's a divine miracle placed by a god who wishes for us to open our minds beyond our narrow materialistic outlook. (People believe in miracles.)
That you are rejecting all these ideas (and many more like them) just shows that you are narrow-minded and prejudiced. Or maybe that poorly-constrained ideas like these can all be rejected once we know enough to realize that all of them have very poor explanatory power and seem not to comport with our understanding of physics that has excellent explanatory power.