Drove part of that route 4-years ago from Flagstaff, with a 2-night stay At Gouldings, Monument Valley and then onward to Moab. Had to pass through Mexican Hat and have passed through there before.
Road conditions on 160 and 163 were good to excellent.
We had been to Monument Valley in 2008, but wanted to take the Grands there. Stayed at Gouldings both times. Things were way different 3-years ago than in 2008. In 2008 we could jump in the 4x4 and drive the trails most anywhere. 3-years ago it was more restrictive. For anyone who hasn't been to Monument Valley, I recommend 1 or 2 full days there.
Have never stopped at Mexican Hat, but have driven through it 3-times. Gotta say the place intrigues me. It is right smack on the San Juan River and the river is pretty wild as it passes by the old town.
Can't help you out with any info about HWY 261.
Joe & Doris
2021 Entegra Reatta XL 40QA (AKA: K9Haus)
2012 Honda Fit