I have driven that road a lot over the years. We lived in Grand Junction for ten years back in the 70s and have visited family there often after we moved east.
The road between Ouray and Silverton is beautiful but not one I would want to drive in my motor home.
There are 5 mph switchbacks and it is very curvy with no guard rails and 500 foot drop-offs.
Could I drive it in my motor home? Yes but I wouldn't enjoy it.
People do it in motor homes pulling toads though so maybe it doesn't bother everyone as much as it does me.
You should be able to miss the snow after the end of May. You could still see light snow but it shouldn't last long.
From Grand Junction we go down through Moab then Cortez to Durango.
The Mesa Verde cliff dwellings on the road between Cortez and Durango are worth seeing and the narrow gauge train ride from Durango to Silverton is also worth taking.