I really do not know why children are not taught logic in schools. (I sound like old Professor Kirk in 'The Lion ,The Witch and The Wardrobe'). There is no logic in saying that God is disproved by the existence of negatives - or even evil. Furthermore, to disprove something is considerably harder than to prove it.
Terrible things could disprove a LOVING God, but not the existence of any God. The first idea is a non sequitur. (To be fair, one is logically entitled to ask whether there might be more than one God.)
There are two major points to the issue:-
Firstly, sin in the world is a logical consequence of freewill. We have minds and are clearly able to make choices for good or ill. However defined, sin is the issue. The Christian claim is that ALL bad things - even natural disasters - are the consequence of *sin. The outsider/seeker/atheist/agnostic cannot come to terms with this point from their existing position and therefore perceive this incorrectly as a weakness in the Christian argument. It is like arguing that there could not possibly be an Eiffel Tower. The person who has been to Paris is then at a considerable advantage over the non-believer who refuses to travel to find out.
Once 'in The Kingdom' the Christian knows a great deal more; principally and empirically, the goodness of God.
Secondly, there is the issue of God's promise "Seek and you will find." This clearly links to the 'Parisian argument'. Nobody has any right to claim the non-existence of God if they have not 'followed all the clues' first.
How many people do you know who have honestly and openly sought God and NOT found Him?
* A quite full definition of sin might be - "Wrongdoing; rebellion against God and that which puts a barrier between Man and God."
