I haven't read the whole thread yet, but I was almost wondering the opposite. What if so-called psychics are successful because they aren't limited by the training and experience detectives have? In the example of the vehicle submerged 500 ft into a lake, the detectives hadn't searched that far because of how improbable the location would be. But, the psychic, not being concerned with probabilities, could have guessed that location simply because it hadn't been searched yet, and the detectives who were initially reluctant to search there were now prompted to keep going despite their training telling them they shouldn't. Usually, psychics get called in as a last resort, so it really could be as simple as looking at what has already been searched and picking the improbable locations...when all the probably locations have been exhausted, that's what you have left. If they get it right, they get lauded as a great psychic, and if they get it wrong, the detectives brush them off as yet another fraud and you never hear about them again.
So, could "psychic detectives" really just be people who think outside the box on investigations? If you brought them in on more cases right from the beginning rather than only on the ones where all the detectives ideas have been exhausted, would they do as well or seem as impressive?
On the other hand, you could be right, that they are good detectives who can put together subtle clues or view things with a fresh perspective rather than following a formula.