I agree with what you have written, but usually when someone says that Christ died for all in a general way, they use a verse like John 3:16 to back it up.

If I understand you correctly this is not the aspect you are talking about. John 3:16 as far as I can tell, is talking about an effectual kind of love that leads to salvation. So, I don't think John 3:16 can legitimately be used to support that argument.