The kid's club on the island is ALWAYS open in time for the 5K runners/walkers.
And given that I have never, when doing that 5K, been off the ship first, there have always been been normal non-5-participating guests in line before me, those people could use the kid's club on-island, just like I could, if my son wanted to get off the ship that early but not run with me. (some people have the experience of the 5K people being first off the ship...I just haven't had that yet in 4 5ks on the island...except for the special, paid-for, Castaway Challenge, got off the ship when it was still dark, 5K...but that one isn't the normal 5K)
Since it's open for the runners, I would assume that it would be open for you!
You drop the kids off there; you don't drop them off on the ship.