I'd say Humboldt is perfect for you if you don't mind the +-4 hour drive on way. The camping there is fabulous - remote, serene, beautiful. There are quite a few good campsites around South Colony Lakes, at almost 12,000 feet. You hike past the earlier sites, along the creek by the old 4wd trailhead, and keep pushing higher even when you get near the lakes, until you get to right around tree line. Crestone Needle right outside your front tent flap is an experience worth driving for. (But be realistic about how cold/windy it could get at that elevation.)
Humboldt itself would be an excellent starter 14er for fit people. From S.Colony, it's the west ridge route. There's no real exposure on the route. Somewhere near the top there's a section where the class 2 trail winds around some boulders, but a lot of folks end up scrambling up the boulders for a fun little bit of easy class 3, still without any serious exposure. At/near the summit you come near the north face, which has a spectacular drop, but you don't have to go any closer to it than you want. And the views from there! The Needle, Crestone Peak, Broken Hand Peak, even Columbia-Kit Carson- Challenger aren't that far away.
The folks promoting 13ers have a point for avoiding crowds while still experiencing spectacular and challenging mountains. But they have to realize what it's like to be from out-of-state - when opportunities to experience the mountains are so rare, it's hard to pass on the big boys. Humboldt requires driving right past a number of other 14ers and other great mountains, but it delivers what you're looking for.
One way to avoid the worst crowds, regardless of the mountain you pick, is to do it on a weekday if your family schedule allows. We've hiked some popular 14ers in prime late August weeks, on weekdays, and sometimes seen very few or even no other hikers. I'm sure you wouldn't be alone at South Colony Lakes that week, but there might not be a lot of other hikers on Humboldt if you can go up on a weekday. (Some campers will be there for Crestone Peak and/or Needle, or just to enjoy the S.Colony Lakes area sans 14ers.)
Yep, it's settled. Humboldt from South Colony Lakes. I can pretty much guarantee you'll fall in love with the place.
A vaincre sans peril, on triomphe sans gloire