The JREF prize is an interesting one. Firstly much of the body of research predates it. Secondly, even if Randi is genuine in his offer, there have been suggestions that he controls the criteria by which it is awarded. Randi does not strike me as open-minded on the issue or unbiased nor is the JREF. To enter into a contest with him would take some courage and the phenomena would have to be reproducible on demand and in a hostile environment. If psi phenomena are possible, there does seem to be evidence that it is adversely affected in such environments. One could say 'well they would say that wouldn't they' - but that would hardly be open-minded. :) I find the 'no ones claimed it' argument a red herring.
What you call anecdotes are also called testimony or evidence. A great deal of this evidence has been investigated independently. Were the investigators fools or 'in on it'? We cannot see what they experienced for ourselves so we can either dismiss their evidence out of hand, as you appear to, or accept it may be genuine. This depends on what assumptions we make about the world we live in. If we assume that psi phenomena and survival are impossible then we must accept there is some error in the investigators deductions if they come up with a paranormal explanation.
If we accept there is a possibility of such phenomena. Then we're into talking about probabilities. If the mundane explanations have been excluded, as far as possible, then the probability of a psi explanation increases. I think probabilities is all we can talk about in such a situation.
As for reincarnation and the transmission of injuries - I don't have an opinion. However in the general sense, just because we do not understand a mechanism does not mean such a mechanism does not exist.