The spillway is kind of "around the corner" from the main dam, so I would guess the latter is probably not "doomed," at least at this point.
A realistic disaster scenario would be if the erosion retreats all the way back to the main spillway gates, or if it undermines the emergency spillway. The berm there would then fail, which holds back maybe 15 or 20 feet of lake altitude. A volume of water equal to that depth times the surface area of the lake would then exit over the hillside, completely uncontrolled. That would be more than enough to cause a serious downstream disaster.
The water flows would be very high (~a million CFS? total guess) which would cause even more severe erosion on that hillside. At that point the main dam might be at risk if the erosion traveled far enough laterally.
Indeed, 13 times more dams than France (who prides itself for 12% hydroelectricity), while USA is only 5 times bigger in population. China has the most number of large dams in the world: More than twice the USA, for 4x population.
