We too like to drive no more than 6 hours a day, and IMO that takes some planning. Figure on averaging about 50 mph on the interstates, substantially less than that on 2-lane roads (this allows for gas/food/potty stops). Speed limits are higher in the west than the east, but depending on what you are camping in, you may need to take it slower. (I tend to drive about 62 mph when towing our TT.) You might find yourself surprised that you're not getting farther in one day if you map it out. (Good suggestion from rexlion, BTW.)
I agree with other posters that Mesa Verde is a good addition to the Grand Canyon; most/all of the southern Utah parks are as well, but a lot will depend on what your family's interests are. The national parks are worth the crowds, IMO. There is no substitute for their scenery - as already mentioned, that's why they became national parks. :)
KOAs and similar parks (don't overlook the Good Sam parks with similar amenities) are a good choice for families with kids. I'd make sure to choose CGs with a pool if possible, especially for the ones you'll use in transit - gives the kids a way to blow off steam and everyone to cool down after a long day in the heat.
Other than Disneyland, Knotts Berry Farm (does it still exist?) and perhaps Universal Studios, I don't see much that kids will enjoy about L.A. It spreads forever and is huge - and the traffic moves fast, bumper to bumper. Ask your kids what they want to see & do; if they help plan the trip, they will be more interested in it.
Don't forget things to see & do along the way. We have found it helps reduce trip fatigue if we spend every 3rd or 4th day "in camp" - not on the road, but staying an extra night in the same CG and either relaxing or seeing some of the local sights. Gives you a chance to catch up on groceries & laundry that way too. :)