Been my experience that Kinsman sees a ton of use (esp. Saturdays) in the summer months. I have been in Dave's position as well, needing to send two hikers along to get tent platforms for groups I was co-leading. I would also say that by 2:00pm on Saturdays it is pretty good bet all platforms will be taken.
At Nauman, I would say it is pretty common for folks to get to the tentsites fairly early in the AM, establish their camp, then day hike over to Pierce, Eisenhower, even Monroe or even the other way to Jackson.
On Saturday, at these two places, the earlier the better.
Peace.