I recently camped at Logging Creek, and have camped at Bowman in the past and drove by Quartz Creek. And I think I agree with the others, Bowman would probably be the first choice, although Quartz Creek looked pretty nice.
For me Bowman has a couple of minor negative points; the road up from Polebridge is a bit long and narrrow, but if you've driven to Kintla in the past it wouldn't be that bad.
Also the campground is in a thick stand of lodgepole, so its a bit dark sometimes, although lots of shade, and there's not much brush, which is a good thing in a way, but it also means that visibility between campsites is pretty open for a lot of the campsites, so sometimes it feels like the neighbors are really close. But these are really minor things, and Bowman doesn't fill up that often so it should be easy to find a campsite away from the neighbors. The roads not bad, unless you're planning to make several trips a day back and forth to Polebridge, but thats a tradeoff, the drive might be annoying, but the pastries at the Mercantile might be worth it!
With the lake, trails and beach there's probably lots more to keep kids entertained at Bowman.
Between Logging Creek and Quartz Creek, I'll try Quartz Creek next time, its a bit closer to Polebridge, and just seemed to be in a nicer setting. Logging Creek is almost right on the road (although there's almost no traffic) and is just kind of open, with not much shade, seemed like it could get hot. Its right beside the creek, there's a Ranger Station just across the road, and the Logging Lake Trailhead is about 200 yds away, but might not be a lot to keep kids occupied?
Quartz Creek is also pretty close to the road, and neither of these have a lake which is a big benefit of Bowman. Quartz or Logging might be easier access if you have a motorhome or long-ish trailer.