if area collision prevents you to go over that ledge, simply adding jump will not fix this "issue". You will jump and still not be able to jump over that ledge. Sorry to say that, but thinking that adding jump just because of this is just short-minded.
In case of you didn't notice, there ARE small ledges you cannot get over AND there are ledges that are exactly at the same height as the one you cannot get over but for those you can cross/climb them.
Actually it would. Adding a jump would make the game calculate a 3rd cordinate instead of just 2. Height. So your characters postion would not only be constantly being checked against collision with static ground and objects, but against their height as well.

And this is Final Fantasy, not WOW or clones alike. no Jump other than emote-auto-jump when walking on a ledge or something that could have been jumped please :/
You're right here though. If the game was wow then it would be successful and we wouldn't be having this discussion.