Read k'peshuto, you have to at least admire the brazenness of the statement מחה פּשׁענו מנגד עינך. We know how bad we are; G-d, You of course know how bad we are, but do us a favor and turn your eyes away and pretend not to notice. Are sins like Schrodinger's cat, i.e. so long as they are not נגד עינך and noticed, they cease to exist?
There are a lot of ways to get rid of sin. First and foremost, there is onesh and yisurim. If you know that when you do wrong you get a smack for it, you are not going to keep doing wrong. However, there are better ways to go about curing the penchant for wrongdoing. Says the Sefas Emes, when a person is really close to G-d, it becomes impossible to sin. Think of your relationship to your wife, to your child, to your parents. Surely you don't intentionally act in a way to harm them, not because of the negative consequences of having to sleep on the couch or your child hiding in their room, but simply because these are people you are close to and therefore do not want to hurt. In that same vein, when you are close to someone, it becomes easier to forgive as well.
When we say מחה פּשׁענו מנגד עינך the letter מ in מנגד is the מ indicating a sibah, a cause. We are asking G-d to forgive not through yisurim, not by punishment, not by demanding other forms of kaparah, but simply because we are נגד עינך, we are close, and when you are close, you don't want to cause harm, you don't want to linger on faults. That should be the motivation behind and the vehicle for our kaparah.