I have ridden over the years most of the suggested routes. Couple of thoughts.
The Katy in June (especially after mid-June) gets quite muggy. However it is an easy ride and the bigger kids could go ride on their own as it is hard to get lost on a rail trail assuming you stay on the trail so they may be less annoyed at having to still around the little ones. There are very little "street detours" on the Katy. Depending on your daily mileage, you may need to do a combination of camping and B&B/Hotel if that matters. While you get to see the river peridically, very little access to it.
Mickelson is nice and scenic but the trail surface can get a bit "gravely" at times. Also, it is relatively short compared to some of the others. You did not say how long you wanted the trip to be. The area off the trail has fantastic scenery but probably not great for a young teen to ride in. If you are driving, you could do the trail then hang around the area and see it by car.
There are tons of trails in MN & WI (some road detours required but doable). Weather is buggy but pleasant nights.
The trails from Cincinnati to Cleveland (or vice versa) is nice but a little "urban" for my taste if I were bring young teens. Weather is good. If you started near Pittsburgh, you could work your way west on trail and some low traffic roads so you could go from Pittsburgh to Cleveland also.
If you are looking at something less hot, think MN, WI, OH, and even Washington/Idaho. We rode a lot of the trails from south of Spokane to Trail of the Hiawatha (great trail!) in Idaho and it was fantastic but as you near the Trail of the Hiawatha, the "trail" is rough in places. You would need a car to drive the "non-trail" segements due to the youngsters, otherwise the riding is fine. Weather was pleasant the the trail was mostly paved. There are some "street detours" but the drivers there are pretty good. Very scenic, good weather, but a tad expensive in places like Coeur d'Alene. Spokane and Coeur d'Alene are nice for adults too.
The Cowboy Trial in Nebraska is too remote and the trail surface sucks at times so you end up riding on US-20 so I would avoid that one. I rode the Silver Comet/Chief Ladiga Trail in May and while it was nice it didn't have much to interest the kids.
Being from Oklahoma (next to Missouri), I would probably head north due to the time of year. June can be hit or miss and if you get rainy weather (major storms in late may through mid-June) followed by temps in mid-90s, the humidity just sucks. Then everyone is complaining.