“Does one have to be first infected and then cured before one is “free of disease”?”
No, but there first has to be a disease. There was no sin or original sin in the garden when Eve was created. All she had was untested innocence and the potential to sin since she was created.
You say "there was no original sin" in Eden, but that is because God made it that way.
Eden is not the natural state of man. It is a blessed, graced state God chose for us. That is what lacking original sin means. It is a blessing.