I would choose either Breck or Steamboat. IMO, Vail isn't nearly as much of a "town" feel that the other two are, especially during the summer.
Personally, I would opt for Steamboat because it is further from the Denver area and thus less crowded. Also, since you're doing 5 days in the Bueva Vista/Salida area, going north to Steamboat will get you into a different part of the state. I could imagine spending 5 days in Steamboat more easily than I could in Breck.
I spent many a summer vacation with my parents in Steamboat and really enjoyed it. The hot spring pool located in town has a great waterslide that kids have a blast in. The more natural Strawberry Park Hot Springs are also awesome. They also have rodeos in the summer which are fun for the whole family.
Steamboat offers a Kids Adventure Club which is similar to a day camp with all sorts of great outdoor activities. There is also an alpine slide (described above), hot air balloon rides, gondola rides, mountain biking, ATV tours, horseback riding, hiking, rafting (if you're not doing that in Buena Vista), and even wagon dinner rides. They also have guest ranches nearby which could be a fun and different type of lodging accomodation for your family.