If I wanted to remove only tamed dogs, how would I do it with EssentialsX? I've tried all the following and none of them work:

/killall dog
/killall wolf
/killall wolf:tamed

I think this is deliberately disabled by EssentialsX, because in most use cases this is not wanted.

In most cases, I would agree, but there should still be an option added for this if it's not possible. Using vanilla Minecraft commands for this purpose is very tedious.

I don't know if the command is still available, but I personally user the /killall tamed
Just notice that it will also kill tames horses and other tamed animals, not only wolves.