The premise of the question is false - though that's more the episode's fault not the poster.
At no point does the episode only state that the nine survivors of the Kodos Massacre are the ONLY ones who can identify him as being Karidan the actor. As the question points out - the photographic records are trivial to pull up on the Enterprise computer along with voice identification.
That is, once someone makes the connection, which understandably an eyewitness is more likely to do.
Lenore is the daughter of Kodos and she is the one doing the killing of the nine witnesses/survivors. To her insane mind this may be analogous to if no one survives a tree falling in the woods did it make a sound? No survivors of the massacre may make it like it never happened or, to her, at least un-prosecutable. Certainly it eliminates the most motivated people that could make the connection. But she is insane as the original question points out.
Leighton is also focused on the nine survivors that includes himself, Kirk, and Riley - but this is in the sense of getting the case reopened as highly motivated individuals. I can not blame anyone in the audience for assuming that their testimony would be involved to identify Kodos.
The main cause of the confusion is Leighton's "There were only eight or nine of us who actually saw Kodos." When in fact he means there were only nine survivors of the execution that Kodos personally performed - and thus saw him doing it. This modification is per Spock's comments which are by definition more accurate. It does not mean somehow only 9 people on a colony of 4000 surviving people ever saw the governor and are still around. But that's an easy misunderstanding to have without the below transcript. And no doubt it is a common misunderstanding - I'm sure I thought that too as a kid.
Note the Tarsus IV colony has 4000 people massacred of the 8000 population. Kodos did this to 50% to extend the food supply. So presumably there are another few thousand people still that could testify that Kodos was the governor and identify him. But those 4000 people were not at the execution and survived somehow - but certainly affected by it.
Honestly I don't think I really appreciated before that not only did Kirk survive the colony he survived the gas chamber, segregated life support, or whatever Kodos was using - at least according to the episode. (I'm not going to dive into what some novel may have done with this story.)
The difference between surviving the colony and surviving "the massacre" is subtle - but ultimately that's the most important thing that defines the nine that saw Kodos.
from transcript at http://www.chakoteya.net/StarTrek/13.htm
KIRK: You mean to tell me you've called me three light years off my course just to accuse an actor of being Kodos?
LEIGHTON: He is Kodos. I'm sure of it.
KIRK: You said you discovered a new food concentrate. What am I supposed to put in my log, that you lied? That you diverted a starship with false information? You're not only in trouble, you've put me in trouble, too.
LEIGHTON: I did it to trap Kodos!
KIRK: Kodos is dead.
LEIGHTON: Is he? Is anyone sure? A body burned beyond recognition?
KIRK: Tom, the authorities closed the book on that case years ago.
LEIGHTON: Then let's reopen it. Jim, four thousand people were butchered.
KIRK: Martha, you tell him.
MARTHA: I can't tell him anything, Jim. He's been like this since the company of actors arrived.
KIRK: Kodos is dead. I'm satisfied of that.
LEIGHTON: Well, I'm not. I remember him. That voice. The bloody thing he did.
(He turns his head to reveal it's not just an eyepatch, but a covering over half his face)
LEIGHTON: Jim, Jim, I need your help. There were only eight or nine of us who actually saw Kodos. I was one, you were another. If he's to be exposed,
KIRK: He's dead.
And later in the episode....
SPOCK: I will continue, Doctor. According to our library banks, it started on the Earth colony of Tarsus Four, when the food supply was attacked by an exotic fungus and largely destroyed. There were over eight thousand colonists and virtually no food. And that was when Governor Kodos seized full power and declared emergency martial law.
MCCOY: I've heard of it.
SPOCK: You may not have heard it all. Kodos began to separate the colonists. Some would live, be rationed whatever food was left. The remainder would be immediately put to death. Apparently he had his own theories of eugenics.
MCCOY: Unfortunately, he wasn't the first.
SPOCK: Perhaps not. But he was certainly among the most ruthless, to decide arbitrarily who would survive and who would not, using his own personal standards, and then to implement his decision without mercy. Children watching their parents die. Whole families destroyed. Over four thousand people. They died quickly, without pain, but they died. Relief arrived, but too late to prevent the executions. And Kodos? There never was a positive identification of his body.
MCCOY: What has Karidian to do with it?
SPOCK: His history begins almost to the day where Kodos disappeared.
MCCOY: You think Jim suspects he's Kodos?
SPOCK: He'd better. There were nine eye witnesses who survived the massacre, who'd actually seen Kodos with their own eyes. Jim Kirk was one of them. With the exception of Riley and Captain Kirk, every other eye witness is dead. And my library computer shows that wherever they were, on Earth, on a colony, or aboard ship, the Karidian Company of Players was somewhere near when they died.
MCCOY: It's unbelievable.
LENORE: No, Father. The time will never come. Tonight, after my performance, the last two who can harm you will be gone.
KARIDIAN: What are you saying?
LENORE: There were nine. Now there are only two, and they will be gone as soon as I. Don't look at me like that.
KARIDIAN: What have you done?
LENORE: What had to be done. They had to be silenced.
KARIDIAN: All of them? All seven? More blood on my hands?