Journey time, plus sight seeing
No matter how hard you try you will miss out on some great places, there are just so many !! Its about finding a balance you are happy with. You say you don't want to spend all the time driving? From day 2 to day 12 you will drive around 2000 miles just to get from A to B which will equate to 4 to 5 hours per day driving with a lunch stop. (For distance comparision from the southern most point in Italy to Oslo in Norway before any detours) Then you will spend a lot more time in the car getting around Yellowstone and other National parks, then you need to get out of the car and actually see things, walk take in the sights and take in the fresh air ! So it's possible if you want a brief glimpse at things and take a few photos and are happy to spend a lot f time in the car.
...and to Save Time and Miles
As others have noted, your overall plan is quite ambitious. They have offered a few places where you can cut back and make your trip better because of doing so. Since I've suggested things to add to your trip, I should also offer places to cut. One would be to substitute Zion National Park for Bryce Canyon. Zion is easier to get to and just as rewarding. Take UT-9 east from I-15 to Springdale UT, visit Zion and then return to I-15 using UT-9/UT-17. You had this in your original plan and I would return to that. It's not a big savings in time or miles, but every bit will help.
I would echo Michael's warning about getting to St.George on the day you arrive in Las Vegas. If nothing else, you could move one of your Las Vegas days from the end of your trip to the beginning. But you will need time at the beginning to get over jet lag and get set to continue at your leisure rather than in haste.
The other thing that people have warned you about and I will reiterate is the fact that travel through Yellowstone National Park will be slow. The park has a very short season and the roads will be packed. It will be standing room only at Old Faithful, and when the road is blocked by bison herds you will not see much or move at all unless you're one of the first few cars in the traffic jam that ensues.
The only other place I see where you can cut without too great a loss is between Moab and Las Vegas on the return portion of your trip. Both Monument Valley and the Grand Canyon (South Rim), despite their undeniable beauty, are scenic drives without a whole lot more to do other than take pictures and short hikes. Seeing both on this leg could be done comfortably in two days rather than the three or four you've got scheduled.
AZBuck