If I take a few steps back I can see some benefits to using them. Having met a few self proclaimed psychics, what I took away from that experience was that they are usually very perceptive and intelligent people that have incredible skills at reading people. This is course sets up the opportunity to scam someone and that is probably where they get their bad reputation. Social engineering can be used for both good and bad. In my opinion the act they put on is just flare to distract from their ability to read people and give them an out when they fail to piece things together.
This is just a theory of mine and I have many wacky theories, but I could see psychics being a human resource for police much like a psychological counselor is to an individual, meaning that the police may already have the data/answers they need but may not be seeing the bigger picture. Perhaps a psychic can observe what police are stating and help guide them into finding answers themselves. The closest analogy I can muster is the fictional Sherlock Holmes using their ability to see all the variables and put them together in a bigger picture when the fictional police could not see the answers that were right under their noses.
Or perhaps I am giving psychics too much credit.
I think it is unusual, if it ever happens, that the police seek out a psychic and pay them to help with an investigation.
I doubt they ever "go to" a psychic. It's probably more like a psychic shows up saying they come with tips. Then they advertise that they helped solve the case. Without the proper training and licensing, they're definitely not on the police payroll.
I'm sure the vast majority of calls to Crime Stoppers are probably people's imaginations gone wild. But without leaving the door open, there is a lot they could miss.
If psychic power really was real and useful, I'd be tempted to actively discredit its existence and exploit its power to finally achieve world domination. I'd be very happy that psychics were banned from police investigation because it would make my crimes against humanity easier to cover up.
Because psychics have their own underground gossip networks like no other.
Maybe a little bit like you would have in the movie "Escape from New York" where everybody knew more than the cops would ever know.
At the end of the day, juries aren't required to give weight to a psychic's testimony and defense attorneys are free to criticize its merits. Just like they can call into question the merits of bitemark and blood splatter analysis, cadaver and drug dogs (and to a much lesser degree, DNA evidence).
Police investigators are not incentivized nor required to hold their investigations to scientific rigor nor are prosecutors to use what "evidence" they find in good faith.
It's a travesty in a lot of ways, but the best defense are attorneys who can tear it apart under scrutiny.
And to close it off, the people who decide the rules of evidence are not scientists. They're lawyers, judges, and politicians. If you're looking for a reason "why" something is allowed in a criminal proceeding look to blame the humans that judge whether or not some practice is as good or bad as another and if it could be presented to a jury. It's not rhyme or reason but human nature.