We go to the Deer Valley YMCA family camp in PA.
http://www.deervalleyymca.org/
It is as you describe-- communal meals, planned activities and free time. In the mornings, kids go off with their counselors for supervised activities while the parents can either have free time or participate in something like a volleyball game, hike, sailing lessons, etc. They have week-long sessions all summer plus weekends -- Memorial Day, Labor Day, and some other fall and spring weekends.
We started going when our kids were 3 & 5, did the full week for a few years and then switched to Labor Day because the week wasn't working with other demands for our vacation time. Kids are teens now and still look forward to our annual trip.
The challenge is that it's hard to get a spot since the return rate for campers is >90%. You'll have the best luck getting off the wait list for the 1st or last weeks of the summer. Which is actually good because those weeks tend to have more new families for that reason.
Also, one caveat that's a problem for some people but not us, is that the camp is alcohol-free.