An Outline and Some "Greatest Hits"
Dave is right that much will depend on the 'dots' you choose and the pace you choose to see them at, so it might help to roughly outline the trip you've already described and see where it gets you to and what that all might entail. I'll look at your loop in clockwise order, although you describe it in counter-(anti-)clockwise order and you can certainly do it either way.
Heading north from Las Vegas, and following US-89 more than I-15, you'd pass within visiting distance of Zion National Park and Bryce Canyon National Park on your way to the Salt Lake area. From there, I'd recommend staying on US-89 all the way up to Grand Tetons National Park and Yellowstone. The section between Yellowstone and Denver is where you'll really get a feel for the 'wide open spaces' of the American West, and this is the section where a close examination of the different possible routes and smaller-scale attractions along them will be required to suit your own needs. Whichever route you choose, aim for Estes Park CO and Rocky Mountain National Park. If Cody WY is on your list for its namesake famous showman, note that his grave site and a small museum is in Golden CO just west of Denver. Next up would be the drive to the Four Corners area and while, again, there are multiple ways you can go, I'd suggest using US-285 to Poncha Springs and then taking US-50 through Black Canyon of the Gunnison National Park and then US-550, the Million Dollar Highway, down to Durango . Next use US-160 west to Mesa Verde National Park, make a short detour north on US-163 to Monument Valley Navajo Tribal Park, and continue on to the Grand Canyon. Finish your loop on I-40 and US-93 over Hoover Dam and back into Las Vegas.
Again, those are just the highlights! All-in-all, that will take about 2500 miles of driving plus any side trips you decide to include. Eminently doable in three weeks, but you will have to apportion your time judiciously.
AZBuck