We have driven the entire Butler Wash side several times and at those times you could do it in any passenger car. There are times during or after storms when it would not work. Some places are a bit rough but just go slow over them. The method I use is to just turn around if there is an impasse.
We have driven parts of the Cottonwood wash side, the west side and had no problems there either but the same applies. They are dirt roads with lots of washes running across them, dry most of the time. Storms can change that in a few hours.
You will see people here and there. Some of the parking places have a few vehicles there, folks going in to see the various sights, pictograph panels and ruins or just the natural beauty. We have been there in October, didn't have any huge crowds but there were scattered vehicles and people. Easy to hike away from though.
Like all of southern Utah, it is more crowded than 10 or 15 years ago but you can still find stellar campsites and places to explore.