I've never actually installed drain tiles on a playground site, but as far as the KaBOOM! build process goes I think it would make sense to install them on Prep Day. If you're concerned about having rocks on site over top of the drain, maybe consider using pea gravel-- this would also drain water through but be considerably less hazardous to have on the site.
But I'm a bit concerned about the process as you've explained it. Maybe I'm misunderstanding, but it seems like you're planning on putting just a thin layer of geo-tex fabric (like landscaping felt?) over the gravel and then putting safety surfacing over that. My concern with this plan is that if the safety surfacing (which can move because it's a loose-fill material) is swept away from over the drain area, there's nothing between a kid's head and that rock or gravel except a thin layer of landscaping felt. Safety surfacing vendors sell a layer of padding that you could buy in a square the size of your drain tiles and put on top of the gravel, under the fabric, that would make this all much safer. Here's an example: http://www.zeager.com/wcduradrain.htm. Actually, this product in particular can actually take the place of the gravel or rock layer entirely, being suitable to lay directly over the drain & allow water through but still provide cushioning for under the safety surfacing. I'm sure your safety surfacing vendor could explain all this in better detail, but I hope this helps.
Happy playground building!