Surely they knew what Kodos looked like. Kirk found a picture of him easily enough and had access to a recording of his voice. Possibly, only the unfortunate 4,000 actually heard and saw Kodos give the orders.
But that's just the thing: everybody already knew everything about Kodos. There were images; there were voice recordings. There probably was a full DNA trace for all we know.
This leaves nothing of value for Kirk, Riley and Leighton to witness - unless they in fact hold information about Kodos that doesn't come from the time of his reign of terror, and somehow differs from the widely available information from that time. That is, they might have information from a time when he was not
Sure, it might be that only the Fabulous Nine would have been witness to the sort of specific utterances that would seal Kodos' fate in a court of law. But nothing in the episode hints at such, and indeed it would seem silly that anybody would need to hear a third party deliver a testimony of Kodos' willingness to execute colonists when the very recording of his voice has him deliver the incriminating orders!
Rather, the episode emphasizes that only the Nine "saw Kodos" or can "identify Kodos" and only Kirk and Riley now can tell for sure whether Karidian is Kodos. Which makes no sense at all, but could at least be claimed to make sense if the seeing and identifying is all about "the man who once called himself Kodos".
Its one of those scripts where they probably needed another rewrite before filming.
The idea that Kodos can walk off a transport and declare himself Governor seems far fetched. The dialog implies Kodos was in charge before the famine happened. Not sure what you mean about Kodos being someone else. Does it matter if he's Kodos, Karidian or John Doe?