at least give the entity being shoved a DC10 Dexterity saving throw (this could represent something like catching the ledge).
+1 Maybe it should be a reaction? This would add some tactical thinking: do I use my reaction for [other ability] or do I save it in case I get shoved? If it becomes a reaction, I'd suggest that an edge-hanging character be un-shovable until their turn so that the enemy can't just shove them again.
I've killed late game bosses with this ability. Given HP isn't blood and how swingy it is in both directions and how many damn chasms are in the game I feel like HP could be used as a buffer against this. Otherwise at least limit the shove action so this isn't so rampant.
What, have anyone who is shoved into a chasm reappear (like flying/swimming creatures do when shoved into a pit/water) but lose 30 HP? ...I don't hate it. It's a band-aid to the problem, sure, but it would add balance to shoving while still allowing those players who love shoving to keep their fun.