Hello Frankie, nice to see you.
I have basically the same problem with this discussion as in a debate about the existence of God. The main issue is not the data (in this case, what children may say, under which circumstances) but its interpretation and the mental constructs this interpretation resorts to and aims at.
Here the central, yet unquestioned and unexplained concept, instead of "God," is "individual". Presumably "human individual". What is an individual? How can it be said to be different from, or the same as, another? What is new and original to a given individual, what is old and inherited? The only "thing" which appears to be new and original is not really some-"thing" but a particular combination of existing "things" (matter and genes and language and mind structures... and stories). Just as all faces in the world are different combinations of the same elements. And that -- which is no-"thing" -- we call "I".
We cannot deal with "it" objectively except through mythical language. From this perspective, "ego" or "subject" are no less mythical than "soul" -- they carry the same risk of treating the no-"thing" as some-"thing". And all "objective" conclusions are condemned because of this.
Subjectively, on the other hand, we speak and express "ourselves" -- our original no-"thing"-ness just as what we have (or, actually, are) in common with "others". We relate, we connect, we communicate, we love, we hate, we "identify" -- with "all" perhaps but always through particular others. A story, an image from the other side of the planet or centuries ago can trigger such a sense of identification -- I am this one, I am that. This proves no-"thing" (objectively) but has to be expressed -- mythically.
As an alternative mythical pattern to "reincarnation" for the (Old-Age ) Western mind I'd suggest the classical Christian notion of communio sanctorum ("communion of the saints").