From Williams I'd bet it's faster anyway to take I-40 to Flagstaff, 89 to 160, 160 to 163, 163 to 191, 191 to I-70, I-70 to Denver. That takes you through Monument Valley, Moab, and I-70 in Colorado. Those roads move very quickly in Arizona, not as sure about Utah/CO, but it's 88 miles shorter and I'd bet it wouldn't take more than 30 minutes longer.
Alternatively, do something involving US 550 and US 50 through Colorado. 50 is gorgeous through there, and 550 is famous.
That's a lot more scenic than going through New Mexico- not much scenery there on I-40- I-25 is kind of nice, but it's still just wide open valley.