Written down before or after initial construction aren't substantially different.
I don't have as much specific knowledge of kites in particular, but based on similar structures elsewhere they almost certainly weren't continuously used and maintained. They would likely be used for one or two seasons, then abandoned for potentially decades at a time as the migrations shifted. When herds eventually returned to the area, people would likely go out and do some restoration ahead of the herds. That would be things like re-excavating the pits, replacing any decayed brush/sticks in the terminal funnels, repairing pit walls, and so on. You can't skip parts or animals may escape and the structure may be covered in all manner of brush or sediment that makes identification difficult on the ground.
It's not necessary to have a plan for the kite, but it would still be pretty useful, especially if it's one that's rarely used so most of the people present might have never seen it.