If you edit the post in the same minute as posting it (or within a minute, I don't know), it doesn't say "Last edit on XX/XX/XXXX XX:XX" at the bottom.

It'd be nice if you could edit a post within 3 minutes, like on Reddit, and it wouldn't say that it was edited.
I often want to edit my post, and I do it as quick as I can, because I notice a small thing I want to change or add, or I notice a typo or a error in the BBCode I used.
I don't want people to know it was edited, because it's still my initial post with a minor change, and it was edited like two minutes after posting it.

Personally I don't mind if it's a couple of minutes longer, stays as it is, or that feature is removed entirely. People edit posts all the time, it's not like it's bad for it to show up.

I tested it and it appears that the edit text doesn't appear only if you edit it within the same minute.
If you edit a post within 5 seconds, but within that 5 seconds it's 3:64 AM instead of 3:63 AM (when you posted it), the edit text shows up.

It sounds like the only reason that there is that window of "up to 60 seconds" is because of the way it's been coded, implying it wasn't intentional.

I don't think there should be a cooldown. The edited feature is there for a reason. If you allow too much time for people to respond, only to have the OP change their message drastically, it would lead to a few confusing / out-of-place messages.

Support. All major sites I frequent that allow editing do this, and I for one appreciate the feature.

While showing an edit was performed isn't -bad-, it sucks when you want to fix a small typo, but then people reading it don't know what you changed, when you didn't change anything of substance.