If you have children, have you ever told them a new rule, only for them the next second later to be doing the same thing you told them not to do? That is what that passage is referring to in simplicity. It is human nature to be rebellious, and to do that which we are told not to do. Now that we know what sin is, there is a realization of the truth and an internal struggle to deal with it. Romans 7 is a difficult chapter. But think of when you may have been given a rule, and that rule invoked in you a response to break it when that opportunity came. Whatever rule that may have been. Or, you were told to not ride your bike on the street when all along that is what you were doing. Now you were told you can't ride your bike on the street, but you want to do it because that is what you did before. My understanding anyways.
Yes of course this happens and I agree with this. Deeper still is the question of why does one tend to rebel in the defense of their intense desire to be their own boss rather than submit and trust in a loving guidance? It is because we feel we must prove ourselves. My view of why the law empowers sin is because we are prideful flesh and sold to sin after this manner of vanity. Any attempt to do good out of that flesh is based upon an ignorance that God is our goodness and that there is nothing good in us that is not God's to begin with.
Consequently, if I endeavour to give charity for the sake of my personal stature with God or my peers, it is not out of the pure love of empathy. If I endeavour to even love God with all my heart mind and soul so that I strain with all my might to keep that commandment by my own volition, I have already failed. But when He reveals Himself to me wherein I cannot help but adore Him for Who He is, I cannot fail to keep that commandment. I therefore don't seek to keep the law anymore. But by the Love that I know is Him inside me, for which I am truly thankful, with all purity and confidence, and with all humility, I trust I will do what He intended in the law. I think you know this from the things you write. In summary, the law empowers sin because when I try to prove I am good, I prove I am bad.
See here the problem with semantics:
Romans 2:13
New International Version (NIV)
13 For it is not those who hear the law who are righteous in God’s sight, but it is those who obey the law who will be declared righteous.
Romans 3:20
New International Version (NIV)
20 Therefore no one will be declared righteous in God’s sight by the works of the law; rather, through the law we become conscious of our sin.