The problem with that plan is that so many aspects of the way the system is designed give people with money and/or time an advantage that you'd basically have to scrap it.
For example, we have a bicameral Legislature and an independent Executive chosen via staggered elections. The Legislators are independent actors. That means policy-making tends to the crowd-sourced-cluster-fuck when things are going well. It also means intricate stratagems of getting Rep A to trade horses with Senator B, while bribing Subcommittee Chair C, etc. become possible. And Bill Gates is the guy who has the time/money/employees do engage in such stratagems. The staggered elections mean that the people in power are looking at vastly different electorates, which in turn means that the guy whose worried about being elected in a non-Presidential year has to worry more about older, whiter, more conservative voters who tend to vote every time; whereas the guy whose next up in a Presidential year is going to be much more concerned with younger, browner, leftier voters who are much more likely to only show up once every four years. If you add in our campaign finance system, and large districts (our smallest House District is a half-million people), it just gets worse.
Compare this to Canada. They have a lot of the same trappings we do like a Senate, but their Senate is toothless. Half the bullshit that allows the wealthy to out-manuever the rest of us is gone because nobody gives two shits what a Senator says. One of their core principles is called "Responsible Government," which means the government is designed so that it's virtually impossible for anything of note to happen without everyone knowing precisely which two to three people to blame if it turns out to be invading-Iraq-level-dumb. See the Commons choose the Executive, the Prime Minister, chooses the Cabinet. If the Commons fail to agree with a PM they will vote against the bill they don't like, forcing a new election, and the next PM will agree with the next Parliament on that particular issue. That means that the only people who can really be blamed for fuck-ups are the PM, the relevant Minister, and possibly (but extremely rarely) somebody else for bullying them. There is very little space in the system for a clever person to game it by clever maneuvers, which means that clever people can't sell access to their clever plans to game the government.
Don't get me wrong. The wealthy will always have more influence then their numbers indicate because a) they vote, and b) many of the not-wealthy figure "a poor man never gave me a job" and out-source their policy preferences to rich-ass-mother-fuckers. But the system we got amplifies that a huge degree.