How about this? Fly to SLC or another western airport, rent a car, and focus on either Yellowstone/GT or Grand Canyon along with Zion & Bryce. We've gone the opposite direction and flown into Raleigh/Durham with our tent and sleeping bags and done the coast down to GA, camping all along the way. With only 10 days, we didn't have the time to drive, but the camping was just as glorious and we had the time to do at least a little exploration along the way.
If you can take the full three weeks and still want to bring your rig, just do YNP and GTNP. Don't try to see much along the way, just beeline there and back. You might get 10 good days in the parks that way.
