In our world, the police going to a spirit medium for the DL-6 case, and being ridiculed might be logical, since spirit channelling isn’t a real thing, but in the world of Ace Attorney, it is.
Not only is it a known and established practice, with detectable physical effects, but the monarchy of at least one country is specifically sought out for their spirit-channelling powers by other governments, so that they can commune with the dead, and receive advice that way.
However, it also seems to be disbelieved, and ridiculed as a pseudoscience, despite that.
The easy answer is because the plot of ace attorney isn’t very serious, and the events happen because the plot demands they do, such as the law that dictates all trials must end in 3 days.
The hard answer (which is never explained but makes some semblance of sense and is my head cannon) is that Redd White controls the media, and has crafted the narrative that mediums are unreliable specifically to create blackmail material against the prosecutor and Misty. It is unknown if the general public believed that spirit mediums are unreliable before the Misty case, but you could construe that not really, and that White has manufactured public opinion in believing that it is.
As a side note, there are plenty of proven techniques that are not viewed as acceptable sources in court in real life too, however here it most likely is because the game doesn’t want you to be frustrated that you couldn’t simply solve all cases by going to a spirit medium.