Lessig is a smart guy, but on this issue he's simply wrong. It's not money that determines who wins elections, it's incumbency and gerrymandering. Why? Because the incumbents already have name recognition, electoral advantage, access to fundraising machinery, and are known quantities to the rent-seekers - they are the safe bets. 95% of the House elections are won by the most money because incumbents have massive fundraising advantages. And they would love nothing more than to further restrict the money spent on elections, because then their other incumbency advantages would count for more. Taxpayer-supported elections would make the problem worse by virtually guaranteeing that any incumbent would stay elected until they die.
Further, public choice mechanics clearly explain why the only way to break that lock is to restrict the money and favors they can direct to their supporters - i.e., cutting at least the parts of government that are the most easily abused, like economic development grants and loans, and disbanding government crony organizations like the Import-Export Bank and the green energy loan programs. In short, you will not remove the corruption without removing the reason for the corruption - the vast power available to the members of Congress and the Executive to reward their supporters and hinder their detractors through government action.