It's an incredible drive. We did in in a 30 Class C with no problems. We went from Bryce to Capitol Reef. So can't comment on the part of Hwy 24 East of Capitol Reef.
Yes, you need to keep an eye on the weather. We had snow at the highest point, saw lightning coming down towards Torrey, and hail when we got into Torrey.
I tried to record some of the drive but ran out of memory in the dashcam. The views are that great.
If you, or your passengers are squeamish about drop-offs or driving next to cliffs, you need to balance that with the incredible scenery. Driving the Hogback, the passengers will get to enjoy the scenery, but whoever is driving will be watching the road.
But my answer is do it! It's worth every minute.