Trade Promotion Authority, a key first step to concluding a Trans-Pacific Partnership trade deal, was filibustered to death Tuesday afternoon in the US Senate by Democrats bucking the Obama administration over its key policy priority for 2015. Recognizing the nature of the uphill battle, over the past week the White House has made the battle over TPP with Elizabeth Warren two-sided, starting with a presidential interview with Matt Bai, in which Obama called Warren "absolutely wrong" about the law's implications for the Dodd-Frank financial reform bill. (For the record, she's partially wrong and partially right.)

But even though Warren has positioned herself in the press as TPP's leading opponent, and her name helps drive clicks to otherwise dreary trade policy stories, Obama's problems securing Democratic votes for his trade agenda have very little to do with her. Indeed, if anything, the focus on Warren as a personality helps cover up the biggest holes in the administration's case, and may help them secure the votes of the crucial bloc of swing Democrats they need.

The coalition against TPP is very broad

Elsewhere in the same interview, Obama observes that "Elizabeth is, you know, a politician like everybody else."

Indeed she is. And like most politicians, wherever she stands, she typically has a coalition of interest groups standing behind her. In the case of TPP, that happens to be a very broad group. It includes the AFL-CIO, whose president, Richard Trumka, told Vox the administration's key argument about TPP being necessary to prevent China from writing the rules of the road is "almost laughable."

But it doesn't stop there.

Elizabeth Warren is an effect, not a cause

When House Democrats talk about the crass politics driving them toward a no vote on TPP, they don't talk about Warren. They talk, first and foremost, about labor unions that have long been the leading lobby against trade deals. Secondarily, they talk about how much backup unions have, from a whole range of interest groups and experts with less political muscle but enough credibility to make people feel comfortable that labor's complaints have merit.

For a Republican president, of course, this wouldn't be a problem.

Big pro-business trade groups like the Chamber of Commerce and the National Association of Manufacturers support the TPP. And siding with the Chamber of Commerce and other business groups against a broad coalition of liberal interests is natural ground for a Republican.

For Obama, the situation is more awkward. It's easy to explain an occasional disagreement with an ally. Maybe the United Auto Workers are opposing something that will be bad for their members but would be broadly beneficial to the economy. But the sheer scope and quantity of groups opposed to TPP makes this a tough sell. So Obama isn't arguing that. Instead he calls TPP "the most progressive trade deal in history," essentially arguing that the whole constellation of progressive groups are making some kind of giant mistake.

Maybe they are, maybe they aren't. But that interest group opposition is what's driving liberal anti-TPP sentiment on the Hill. Warren is against TPP for the same reason everyone else is — the groups she's allied with are against it. To see how true that is, look at something like Oregon Senator Ron Wyden's last-minute decision to join with other Democrats in blocking a crucial pro-TPP vote. Wyden's an idiosyncratic, wonkish senator known for dealmaking more than for headline-grabbing fights, and he's not, by any means, a Warren acolyte. But a mix of pressure and arguments from groups he's allied with have brought him around.

The Obama-Warren fight gives some Democrats cover

One House aide whose boss is staunchly anti-TPP explained to me that while Warren isn't driving opposition to the deal in the Democratic caucus, fighting with her might help win the deal some votes. Going negative on Warren is a way "to provide some cover to the New Dem types" who are happy to buck liberal groups, but don't want to leave themselves too vulnerable to possible problems with their base.

Obama and Warren are both popular with rank-and-file liberals, so a good high-profile Obama-Warren fight serves to somewhat muddy the waters. A look at the interest group coalitions around TPP, by contrast, would reveal a pretty classic left-right partisan spat.