We did a southwest road trip this past June. We made it to Amarillo, TX, the first night, then we headed west to Flagstaff with a stop at the Petrified Forest. In Flagstaff we stayed at the Drury Inn on the college campus and I give it high marks. Our room was a suite and it was huge.
For Zion, we stayed in the Best Western that was right outside the gate. It was a bit pricey, but it was very clean, had a nice breakfast, and we could get to Zion easily at night for stargazing. Bryce was just a daytrip on the way to Moab and in Moab we stayed in another Best Western (also nice).
No help with Yellowstone, but I do think it's quite a drive. There's so much to explore in southern Utah that I would do GC, Zion, Bryce, Arches, Canyonlands and some state parks instead of heading up to yellowstone.
We did over 4000 miles going from MO to CA, but most of our driving days were 3-4 hours.