We've done the Salt River Canyon climb in and out of the canyon a couple or more times, despite being warned not to do it. (the last time about 4 years ago when we met Jim & Tarry Johnson at Springerville.) We don't remember it being any more of a problem than the the Escalante Staircase in Utah. There is a third lane for slow drivers going up, and we just took our time. It is a gorgeous drive and there were several pull-outs as I remember it. Near the Salt River Canyon are a bunch of fairly steep switchbacks, but we took our time and had no problems. You can make your own decisions. We've been on a few roads around the country, that we really shouldn't have been, but neither of us remember Hwy 60 being one of them, nor did we make note of it on our maps. I guess you should know your own comfort zone for squiggly roads, so you're on your own with this decision.
However, I would echo Paul in that it is not a road for night driving. You really need to know or be able to see your way on it.
P. s. We were driving a 38 ft. mh with Jeep tow.