An emergency spillway is good news, but not a guarantee. If the walls of the spillway get flanked, well, as you point out, it is an earthen dam. Nothing to panic over right now, but something that needs to be watched.
I see Livingston handled 110,000 cfs in 1994.
Thats a LOT of water. So it can surely handle this.
Especially if the amount of water accumulating behind the dam from the rain, exceeds the capacity of the spillway.