I rewatched this episode recently. It's quite entertaining (in fact Ron Moore cited it as his favorite OS episode), though it's not terribly subtle in its allusions to Hamlet, and the two main guest stars chew the scenery like a stick of Juicy Fruit.
There's a couple of confusing things I came away with.
1) Obviously, this episode was produced in a time before things like DNA identification or facial recognition technology were commonplace, but you now have to wonder what the "in-universe" explanation why such things aren't used.
2) Kirk's determination and his ethically dubious actions would be repeated in "Obsession". Much like in that later episode, he keeps his reasons from everyone, until Spock and McCoy confront him on it.
3) Obviously, the episode was supposed to imply that Leighton was scarred on Tarsus IV, though they're clearly ambiguous about how that happened. I also find it hard to believe that reconstructive surgery hasn't advanced to the point that he wouldn't need to cover half his head with this
4) Oddly enough, no one really brings up the idea of Karidian/Kodos being put on trial. It's rather strange that McCoy would ask Kirk if he'll "play God" and "carry his head through the corridors in triumph". I don't believe for a second that Kirk would ever actually kill Karidian, no matter how sure he was of his guilt. Surely turning him over to the proper authorities is a simple enough matter.
5) The "phaser on overload" scene was fairly well done, but is it believable that would randomly have these waste disposal chutes in corridors that let you toss any old garbage or unwanted items into space (presumably after passing through some airlock, of course). It's also unsettling to think that a civilian like Lenore could steal a phaser and sneak into the captain's quarters without anyone noticing. (Seems to me it would be better off if Kirk left his security officers on the ship, instead of bringing them on away missions to die untimely deaths.)
6) This is a minor quibble, but I would've like to have seen some kind of reaction from Riley after Kodos was killed.
7) This is a big one: What exactly is so special about being one of the nine eyewitnesses anyway? Why couldn't any of the other ~4000 surviving colonists identify Kodos? Are we supposed to believe he was some sort of Howard Hughes-type recluse who never let anyone see his face? That doesn't seem like the kind of trait that would lead to someone becoming a governor.
And, more importantly, this all seems rather moot, since they had a picture of Kodos in the database! So, again, what's so special about being one of the "eyewitnesses"?
Still, as I said, an interesting episode.
There's a couple of confusing things I came away with.
1) Obviously, this episode was produced in a time before things like DNA identification or facial recognition technology were commonplace, but you now have to wonder what the "in-universe" explanation why such things aren't used.
2) Kirk's determination and his ethically dubious actions would be repeated in "Obsession". Much like in that later episode, he keeps his reasons from everyone, until Spock and McCoy confront him on it.
3) Obviously, the episode was supposed to imply that Leighton was scarred on Tarsus IV, though they're clearly ambiguous about how that happened. I also find it hard to believe that reconstructive surgery hasn't advanced to the point that he wouldn't need to cover half his head with this
4) Oddly enough, no one really brings up the idea of Karidian/Kodos being put on trial. It's rather strange that McCoy would ask Kirk if he'll "play God" and "carry his head through the corridors in triumph". I don't believe for a second that Kirk would ever actually kill Karidian, no matter how sure he was of his guilt. Surely turning him over to the proper authorities is a simple enough matter.
5) The "phaser on overload" scene was fairly well done, but is it believable that would randomly have these waste disposal chutes in corridors that let you toss any old garbage or unwanted items into space (presumably after passing through some airlock, of course). It's also unsettling to think that a civilian like Lenore could steal a phaser and sneak into the captain's quarters without anyone noticing. (Seems to me it would be better off if Kirk left his security officers on the ship, instead of bringing them on away missions to die untimely deaths.)

6) This is a minor quibble, but I would've like to have seen some kind of reaction from Riley after Kodos was killed.
7) This is a big one: What exactly is so special about being one of the nine eyewitnesses anyway? Why couldn't any of the other ~4000 surviving colonists identify Kodos? Are we supposed to believe he was some sort of Howard Hughes-type recluse who never let anyone see his face? That doesn't seem like the kind of trait that would lead to someone becoming a governor.
And, more importantly, this all seems rather moot, since they had a picture of Kodos in the database! So, again, what's so special about being one of the "eyewitnesses"?
Still, as I said, an interesting episode.