According to scripture that was wiped as you say for those who love the Lord for that sacrifice and believe on him. It doesn't clear those who do not love him nor believe.
Children are born in sin as David said but it is not held against them while they are too young to know right from wrong or what scripture shows to be an age of accountability. During this time they sin like some adults and have illnesses.
Once a child grows to know right from wrong they can either believe or not believe on the Son of God. Then they are accountable and subject to the results of their choice to either believe or not believe on the Son of God.
Healings and deliverances and such are all areas too lengthily to go into however as they encompass to many scriptures and life events and such.
So, yes, when Jesus died for the sins of man he wiped the slate clean for all who love him and believe on him.