So typically, in the U.S., the investigators (police) will give the case to a prosecutor who will review the case and the evidence and decide if they will present the case to a judge and if so, what will be charged. The prosecutor is the ultimate authority of how to present the case to the court. The investigator(s) will likely be called to testify as their expertise's and actions that lead to certain evidence is required. If there is a certain expert in some type of evidence, that person may testify on conclusions of their findings (for example, if the coroner's report concluded that some was the victim of a homicide, the coroner who wrote the report will be called in to testify to his findings... but since the coroner did not have access to the Detective's witness, they wouldn't talk about the evidence with respect to that which came from eye-witness statements.
If you're basing you question off of TV or Film depictions of Court rooms, keep in mind that many court room scenes are not accurate and the investigator presenting all of the case could be a result of a limited cast, budget, or time for specialists to present the case. Some shows explicitly separate the roles (Law and Order and its spin offs are a one-hour shows, with half an hour being police procedural and the second half being a legal drama. CSI and its spin-offs would frequently bundle the CSI field workers as investigators for storytelling purposes (in the original show, most of the cast were not detectives and did not interrogate witnesses. Jim Brass was specifically the detective who affected the arrests, often having the CSIs explain the science. The show also never got to the trial phase, and in some cases, what if any crime was committed is never stated, as the story focused on getting to "how and why something happened" not the legal ramifications. In one episode, Grissom explicitly tells someone he doesn't have enough evidence to prove the fraud in criminal court... but the insurance companies will see his report and it's enough that they would be able to refuse to pay.
In shows that are not explicitly legal procedurals, it might be that the cast of characters is not sufficient to cast for all the witnesses AND/Or the characters are not in professions where they would know how to properly run a court. For example, in one episode of the Brady Bunch, the Brady kids, all of them are High School age or less, hold an ad hoc trial with Alice (the family's maid) as the judge. The case descends into childish bickering and Alice calls a recess when she realizes that her Roast is burning.
