Eyewitnesses are still an important part of the judging system. It probably had an even bigger role in the past. That were the times before Purkinje, Overton and Herschel made fingerprinting a normality. That was before Mullis and Jeffreys discovered what is needed to use the DNA in the criminal cases. Before the time of mostly cheap but even more importantly ubiquitous recording devices like cameras. Before we could see someone history by reading their email archive.
The judging system in most part works in a way that the better side wins. Not the one that was more right, or the one where it would be more fair, but the one that manages to convince others, depending on the system either a judge or a jury.
The eyewitnesses are the same way. They are obliged to tell the truth, but they need to convince people that they are right. It a way it could almost be described as a popularity contest. Do you know one thing that doesn't really work that is based on the similar popularity metrics? Politics.
There are some ways of trying to make the eyewitnesses more scientific. One way of how they are trying to do it in New Jersey is by telling people about the different scientific research that gives rise to the problems with record of eyewitness. Like how memory is not a video recording (which for the record, I did not know anybody still thinks that). The effect of the leading questions and so on.
But Papailiou, Yokum and Robertson showed in their research that this does not make them more likely to distinguish between a good or a bad report. Instead they became more skeptical of everything.
Which is only partly good. It is good for the legal system of 'innocent until proven guilty'. Not so good for the scientific method, as it is the admittance of ignorance, not skepticism that is one of the main drivers.
Not to mention, in the criminal law, now by presenting these rules you can make anybody more likely to not be accused of the crime. I find it hard to believe that nobody would take that possibility.
One thing that comes to mind is the legal system in the original Star Trek. There were a couple of episodes with the legal theme. Let me talk about the episode in the 3th season called Turnabout Intruder.
In that episode the minds of Captain Kirk and his former lower Janice are exchanged. So Kirk ends up win the Janice body and Janice is in the Kirk's body. Now Kirk needs to convince others that it is him in the woman's body.
I like the exchange after the mind meld in this scene. It is sad, that they cut the next line: "However, my belief is not acceptable evidence. Evidence must be factual. Doctor McCoy may be of help. Come with me." 
Even if he manages to convince his crew, as long as he is unable to produce any evidence, that is not his own testimony, it will not be counted as such. Kirk first tries to make a claim. Spock says it has to be 'substantiated by any external evidence or objective tests' . Kirk then tired to convince him by offering the testimony of the common past instances, but it is not excepted by Spock.
Even when Spock tests him telepathically, it convinces him, but he says that it not not enough. Scott later confirm this, as he says that telepathic connection is subjective and not something that can be examined in the open.
Even when everybody start to doubt that whoever is in Kirk body is really Kirk, they don't fight through the judging system, but, as clearly seen in the case of Chekov and Sulu, through civil disobedience.
I want to talk about the upper episode through the hand of the conditions for science. There are three conditions for science:
- the admittance of ignorance
- the equal access to knowledge
- content thinking
The admittance of ignorance means, that we are willing to admit that there might be something we don't know. It shows in McCoy's admitting that something might be up even though his medical scans show nothing. It shows in Spock willing to mind meld with what was by his knowledge at the time a woman with mental problems.
The equal access to knowledge refers to the ability that everybody is able to go and confirm or dis-confirm the fact of somebody else. Nothing that some person can check but others can't is considered knowledge. It is why Scott opposes Spock's telepathic proof. As telepathy is not something that everybody can check for themselves it is not considered to be proof.
The third is content thinking. It refer to the thinking, where the raw data is more important than any kind of authority that this person might have. Spock equals it to violence, when he reprimand Janice in Kirk's body that it is uncalled for. He later stands against her, when he claims that she is dangerously insane or when he tired to convince him to stand down. Chekov and Scott might use a more 'subjective' evidence of captain character, but they also don't just take the words of authority as given.
Which is surprising when you think about it. Starfleet is after all a organization with military structure.
So in the one possible utopian future (the only utopian future seen so far that I actually liked), they still have eyewitnesses. But they don't accept their report without any other evidence. It would be an interesting concept to test in real world.