Lessig is a bit of a mystery to me. The obvious solution to corruption is lessening the power of the government. There is no way to remove incentive to steal when the money is so abundant.
I am sympathetic to your point -- power corrupts, therefore let us limit the amount of power -- but it's stil not the whole story. Because corruption itself produces power imbalances.
By corruption, Lessig does not mean simple bribery. He means the corruption of institutional purpose.
An example would be how current American governments engage in gerrymandering with impunity, and are placing reliable partisans in positions where they would be judging the validity of elections. There isn't really any imaginable government that doesn't control the electoral process in this way, so limiting the power of government isn't the answer.
It's like it's not even on his radar.