According to Google Maps: (Phoenix to Yellowstone)
Hwy 89N to I-15 is 1,028 miles; 15hrs. 40 mins.
Takes you by Bryce Canyon, but yeah, I-15 leaves little to see.
Hwy 89N to 160 to 191 to I-15 is 1,068 miles; 16hrs. 36 mins
Takes you by Monument Valley, Goosnecks SP (on San Juan River - spectacular views but boondock only), Moab, Canyon Lands, and Arches.
Hwy 60E to 191 is 1,158 miles; 20hrs. 9 mins.
Takes you by Monument Valley, Goosnecks SP (on San Juan River - spectacular views but boondock only), Moab, Canyon Lands, and Arches.
Plus you'll have a much more scenic (hills and turns) for better views and pass through Jackson WY on your way into Southern Yellowstone.
As you can see, when you stay off main roads the time increases to a much greater degree than miles. The benefit is more so see. The problem is you only have 3 days.
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