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.
I love the way you put this.
I think we can agree that no-one's proved this, but there is interesting anecdotal evidence. Like there was with acupuncture - but then that got proved to any reasonable level of scientific satisfaction. It took thousands of years, but then it's not something that has always or universally been practised. Homoeopathic medicine is still to satisfy a reasonable level of proof, but all the signs are it will.
Whether this thing you're discussing will follow acupuncture in to provability remains to be seen but there's at least one major logical problem - it's unlikely this is a new thing (if it is real). Why then hasn't it been satisfactorily documented on numerous occasions. It's also unlikely to be a geographically limited phenomenon. And pretty unlikely to be a cultural or religiously unique thing (if it is real) - unless there are many god and each set of believers have a different afterlife (or as Pratchett amusingly speculates, people get the afterlife they expect). If it is real, it's rather unlikely to STILL defy a reasonable standard of proof as it is probably been happening a VERY long time (if not as long as since humans were humans) wherever humans were in all cultures no matter what they believed.
It's like mind powers; unless they are an emergent characteristic not universally expressed in the population that hasn't had sufficient time to spread through the population or is not a trait that increases fitness (as measured by numbers of descendent's), mind powers have been around long enough all over the place to make lack of proof suspicious.
I have to admit I would LOVE mind powers to be true, as I would love reincarnation to be true. My fiancé has clear recollections of having met me in a previous life. I have a memory of a few seconds that seems to be real (not a dream memory, too many details) but would have been impossible for me to experience as it would have been hundreds of years ago. I can 'heal' my fiancé by 'laying on hands' on small injuries or hurts, at least to the extent of reducing or preventing swelling and reducing the level of pain experienced. I've done it to another person once - but I was doing it to see if I could make her feel better not because I think I have special power. Although that would be cool.
Despite all this the fat lady still has to sing. Bad science (even if well intentioned) is one explanation for people finding this credible when they read seemingly impressive accounts. I wonder if Randi has looked at claims like this. There's classic examples of paranormal investigators getting great results, but the methodology will turn out to be flawed. They miss things, or fail to exclude un-paranormal explanations or 'contamination' of result. Memories aren't necessarily real, and fabricated (as in unconsciously fabricating a memory) memories often follow archetypes. Little girls saying they chose mummies tummy could have seen this idea on TV or heard it in a story - I've heard actually it on TV before as a sweet explanation given to a little child of how they got in mummies tummy ('you were in heaven and came down to be born and chose me to be your mummy').
And 'healing'; well we already know how powerful psychosomatic factors are in healing - even to the extent of injuries healing faster and pain lessening. My fiancé believes in me; the friend I did it to before is, shall we say, a little suggestible. If you think you're gonna feel better, you often do.
I've tried to OOBE, but would need to set a camera up at the target and have someone DO something at the target to be able to determine if what I 'see' when I try to OOBE is real. I can fly over places I know and see details and people and cars moving below, starting from where I am, and eventually my concentration will go and I lose the thread; but my imagination is easily good enough to make it all up. I can play entire Beatles albums in my head - why isn't it equally possible to recreate a simulacrum in your head of places you know well and imagine flying over them.
I have an emotional involvement that makes me want such things to be real, but I'll not change my standards of proof because of that. Been there, done that, not doing it again.
All very interesting; rest assured 'if the fat lady sings' I'll be delighted. Until then I am sceptical saffron and curious orange.