Kirk states that they are at the planet to find some trace of the starship Archon that disappeared 100 years ago. Later on, once they have get their missing man back, Kirk refuses to leave because he says, "we still haven't found out what happened to the Archon". If the Archon disappeared 100 years ago, what exactly were they hoping to find at this time? After 100 years, all the crew would likely be dead (unless it was a Vulcan ship, which it wasn't). Also, since Landru pulled the Archon into the atmosphere, it would have burned up so there would be no trace of the ship either.