First of all, I did enjoy watching Cannon; I'm old enough to remember it when it was originally on.
Secondly, Kirk did all kinds of things that went against the social taboos of the 23rd century - because to him, they were the right thing to do
Thirdly, it's obvious that for story purposes, Chekov had to "remember" Khan even though there is NO evidence that he was on the Enterprise at the exact time that the events of "Space Seed" took place. That said, it wasn't the planet's official name/designation that jogged his memory... it was the title of a book: Botany Bay.
After he made the connection between the book, the old sleeper ship, and Khan, that's
he told Terrell they had to get out of there NOW. Of course, it was too late by that time.