The palace intrigue of Washington rarely has a direct bearing on the lives of ordinary Americans. There is, though, one elite scrimmage that is worth paying attention to: the fight for the next chairman of the Federal Reserve.

Ben Bernanke, the current chair, is leaving when his term ends in January. The choice of his successor will tell us a lot about the future health of the financial system, the economy and the role of Wall Street over the next few years.

Bernanke was the accidental architect of financial bailouts and unusual market stimulus that, by all accounts, saved the financial system after its own impulse towards self-destruction nearly destroyed the entire economy.

His successor will have the same job: to keep Wall Street from destroying itself and us. It will have to be someone who can listen to Wall Street, but does not do its bidding. The Fed is, in its own description, "an independent government agency that is also accountable to Congress and the public". The words "Wall Street" are nowhere in its mission.

One of the candidates to lead the Fed is Larry Summers. Summers is the brilliant 58-year-old president emeritus of Harvard University and former Clinton-era Treasury Secretary who advised President Obama during his first administration. He has been a reliable candidate for powerful financial jobs – like president of the World Bank – ever since, but he really wants the job of running the Fed. He started lobbying over five years ago.

In 2009, he complained to President Obama and Rahm Emanuel when he was not nominated to the Fed; he also asked for "a seat at the table at cabinet meetings, a chauffered car and Secret Service protection," according to Bloomberg. Summers later parlayed his Fed rejection into the very rare privilege of being allowed to email Obama directly on the presidential BlackBerry.

Here is the problem: Summers lacks distance from Wall Street. In fact, as we speak, he is paid servant to Wall Street. He has drawn millions of dollars in pay from Wall Street firms including Citigroup and hedge fund D E Shaw, which paid him, in 2008, roughly $100,000 a week. He has received five-figure speaking fees from Wall Street banks including JP Morgan.

Currently, Summers' consulting portfolio of clients includes stock exchange Nasdaq OMX, venture-capital firm Andreessen Horowitz and asset-management firm Alliance Partners, according to the Wall Street Journal. That may have made Summers rich, but it's also a lot of baggage for a Fed chairman, who makes controversial decisions every few months about the future of the financial system.

Summers, if he were head of the Fed, would end up turning every single decision – about interest rates, about stimulus – away from a discussion about the country's future, and into a referendum on his own resume and the cornucopia of potential conflicts it presents. He wouldn't do this intentionally; his very presence would be a provocation. Every time he raised interest rates, critics would question whether he was selling out to Wall Street.

That sounds exhausting – for Summers, for the citizenry and pundit class, and particularly, for an already weakened president who can't afford to open one more flank to attacks from his enemies.

Obama already has a very weak record on the economy considering that 12 million people are unemployed and the president's stalemate with Congress has killed any chance of useful legislation. Obama is now facing the 2014 mid-term elections and a fight to boost his party's standing in Congress. He can't wait for Republicans to destroy themselves; he doesn't have that luxury.

Enter into this stew of potential presidential failure Larry Summers. Just by showing up, Summers would, without intention, infect the Federal Reserve's mission with questions about his old relationships and conflicts as a paid advocate for Wall Street's interests.

Summers, despite his Democratic allegiance, would also help erode Obama's base of liberals even further. Summers has a history that drives liberals bonkers because he helped kill Glass-Steagall. (This is true, but lacks context: by the time Glass-Steagall was struck down, it was practically dead anyway. Commercial banks like Bank of America and Chase openly owned investment banks and speculated in everything from stocks to currencies to commodities. Summers and his allies changed a zombie law.) Republicans are not particularly hot on Summers either. He's not their man, and he provides a convenient way to attack the president.

There is also a viable alternative to Summers in the form of Janet Yellen, the equally brilliant 66-year-old vice chair of the Fed who headed President Clinton's Council of Economic Advisers and has spent the years since as a professor and at the Fed. Yellen has been one of the top Fed officials to advocate for transparency. She is the "establishment candidate" from within the Fed.

The battle between Yellen and Summers is really as nasty as economists are likely to get. There are a litany of reasons that people have opposed Summers – and these are prominent, credible people, not the usual peanut-gallery scrum of cranks and conspiracy theorists. They include Senate Democrats and an apparent majority of prominent economists. Their opposition is notable for its loudness. One virtue of this argument is that there is no backdoor dealing; people are openly taking sides in Camp Yellen or Camp Summers.

The arguments against Summers include his perceived sexism, stemming from is controversial comments as president of Harvard about women's capabilities in math and science. This is not a strong argument.

The battle of the sexes here – that Yellen is being diminished because "she is not part of the fraternity. Indeed, she is reminiscent of other accomplished women with whom Mr Summers, or his supporters, or both have tangled in the past," is all too easy. But it distracts from the substance of what we should be paying attention to in this battle.

There's a strain of thinking that goes that Summers has a lock on the job because he has visited the White House dozens of times, and Obama knows him well. This may be true. But if it is, Obama should consider carefully his method of choosing advisers by his familiarity with them.

It has not been, by the evidence at least, a particularly successful one. Nearing the end of the productive part of his second term, the president is still in an unceasing war with Congress, with an undistinguished economic record, and a low approval rating. Maybe it would be a good idea to shake things up.