The entire premise is that the Addams Family are people who enjoy things that most people wouldn't enjoy, namely, extreme torture and pain, poisoning, impaling, executions, etc.

At the same time, when Wednesday finds out the guy she likes is a "monster", she doesn't want to stay with him because he's killing people. But ... everything the Addams Family does kills people if applied to someone other than an Addams Family member. Wednesday was expelled from her regular school at the start of the series for throwing piranhas (which don't really eat people from what I understand) into the swimming pool, presumably so the fish could kill and eat the students she didn't like. What is the distinction? What makes Wednesday different?

The entire premise breaks down. Either you have the Addams Family enjoying gruesome things and seeing the world as different, or you have Wednesday recoil from gruesome things. You can't have both unless you make the distinction clear.