As we have so little documentation, I think that claim needs support.

See: Peter Popoff, Benny Hinn, Jimmy Lee Farnsworth :), psychics, new age "healers", etc.

the supernatural experience is uncommon - whereas at least the small studies that have attempted to collate information suggest otherwise.

I object. Supernatural experience is uncommon. Supernatural claims are rare, even if they can aggregate into a cumulatively large(ish) number. I doubt we would argue that humans have a great deal of experience with extraterrestrial intelligence, even if we have a cumulatively large number of extraterrestrial claims.

If a large number of claims makes something common, then we would have to accept that psychic powers, ghosts, extraterrestrials, a wide variety of gods and all manner of magic are common human experiences.

After all, even if faced with something that might be miraculous, you could simply insert the Science of the Gaps to escape the conclusion

Indeed. But this illustrates the problem with trying to prove a negative claim.

Keener produces some references from people who I'd suggest might be reliable (doctors for example)

A doctor certainly does have some degree of relevant expertise, but I don't think their word (which usually amounts to hearsay about what they supposedly said) is sufficient evidence. "I cannot explain it" does not necessarily mean "therefore, it was supernatural." And people tend to describe extremely rare things as miracles, when they are simply rare natural occurrences.

You are right that we all have different levels of confidence at which we evaluate different claims, and I agree that supernatural claims are not exactly reproducible in a controllable setting. However, it is difficult to see what separates them from a wide variety of other magical or implausible claims that we have little/no trouble dismissing as unjustified by the claims.

Also, whoever is downvoting, at least join in the conversation.

I agree. Please don't downvote Flubb's thorough, thoughtful comments. This is exactly the kind of dialogue that should be encouraged in this sub.