Last night, I brought this up when I finally continued my FFVI livestream. So who happened to fall into the trench and decide that a door needed to be installed in a cave? At the time I said it was a natural forming cave, but Vivi22 pointed out in the chatbox that it may have been a sunken structure of some sort. Even if that were the case, the at door is in way too pristine of condition, so someone had to have gone through the trench and decided to install a door.
This is very important to me.
EDIT: This is by far my most successful thread ever.