In the last 2 years, I've done a 10,000 km trip and an 11,000 km trip. Both in right about 6 weeks.
Adrenaline had us moving on the first trip because it was a new experience. We didn't blink an eye at the driving. But this last one was different. I think the newness had worn off. The driving still wasn't a huge deal, but I was aware of it. I don't think my family would do another that far anytime soon.
As said above, your trip is covering what essentially would be 3 different trips...even for hard core road trippers.
Personally, I wouldn't drive that far for ONLY the Grand Canyon. You would be right next to Utah's Grand Circle... You just can't be that close to them without going (and your time limit wouldn't allow it).
What's in Texas? You're paying the price with a lot of additional miles...
What would I do? Badlands, Black Hills, Yellowstone and Tetons. Or, pass everything until Utah's Grand Circle, including Grand Canyon. Both of those options would still leave no relaxing time.