In Sunday School a child might say, "Teacher what you are describing is a squirrel, but since this is Sunday School, the answer must be Jesus."
You are absolutely right. The only answer for this hand soap is Jesus. You cannot wash your own sins away, you cannot scrub really hard to make the sins fall off. Jesus can and does wash you clean and remove the guilt.
Don't let this false advertising suck you in - Jesus is the answer.
Now, having said all that, how do you "rub your hands together religiously?"