Jim, //if something contradicts where we are at the moment then there is a problem.//
If there’s a problem, perhaps science should take it more seriously and attempt to solve it.
//I am rejecting it -- or am at least highly sceptical -- because there is no reliable evidence. Why is that such a problem?//
What you are saying is that because the millions who have experienced this odd phenomena can only offer anecdotal (and, at the moment, it can only be anecdotal) evidence, that evidence is not worth considering. Whilst many of these people may be liars, or delusional, or potty, many of them are not. There are things happening in this world that we can't currently explain, but because hard evidence cannot be provided, science files it all under the heading 'Nonsense'. That is the problem.