At their convention in Charlotte last week, Democrats seized on President’s Barack Obama’s rare advantage over Mitt Romney on national security, portraying the Republican candidate and his running mate as bumbling foreign-policy novices, repeatedly referencing the president’s involvement in the raid that killed Osama bin Laden, and emphatically staking the Democratic Party’s claim to leadership on national security.

“In a world of new threats and new challenges, you can choose leadership that has been tested and proven,” Obama declared, while U.S. Sen. John Kerry, D-Mass., praised the president for giving “new life and truth to America’s indispensable role in the world” and never asking “other men and women to fight a war without a plan to win the peace.” No wonder this week’s headlines herald a full-fledged Democratic offensive against the Romney campaign on foreign policy.

But here’s a counterfactual: Would Obama be in as strong a position today – with substantial leads against Romney on counterterrorism, foreign policy, and national security – if he hadn’t ordered the raid that killed bin Laden? The evidence is mixed.

To investigate the question, we need to travel back to the days before the bin Laden operation in May 2011. By April of that year, many of the cornerstones of Obama’s foreign policy were in place. He had already ramped up drone strikes against suspected terrorists, ordered a troop surge in Afghanistan, inked a nuclear arms reduction treaty with Russia, and spearheaded the multilateral military intervention in Libya (Obama wouldn’t withdraw the final U.S. troops from Iraq until the end of the year).

And yet, as the operation in Libya got under way, Rasmussen Reports recorded record-low approval for Obama’s handling of national security, with only 37 percent of respondents supporting his policies, down from 45 percent a year earlier.

In the weeks after the bin Laden raid, however, Obama’s national security ratings rebounded and his overall approval rating soared. “Obama’s performance on national security and international affairs and his image as a strong leader appear to be behind his rising approval rating,” CNN observed at the time.

But the bin Laden bounce is a bit misleading. While AP-GfK polling suggests that support for Obama’s counterterrorism policies benefitted from the raid, with approval ratings peaking at 72 percent in May 2011 and hovering in the low 60s ever since, CBS News polling on support for Obama’s handling of foreign policy doesn’t indicate a similar trend (the president’s approval rating has danced around in the 40s since 2010).

What’s more, it’s Republicans who seem to appreciate the bin Laden operation the most. When CNN recorded the surge in Obama’s approval rating in May 2011, it noted that the al Qaida leader’s death had primarily boosted Obama’s support among Republicans and senior citizens. Last December, the most popular answer (27 percent) to a Wall Street Journal/NBC News question about the Obama administration’s “most positive accomplishment” was the killing of bin Laden. But only 14 percent of Democrats chose that answer compared with nearly half of Republican respondents.

Poll after poll this year suggests that voters care about jobs and the economy – not foreign policy. But as The Washington Post’s Chris Cillizza and Aaron Blake noted last year, these results can be red herrings:

“Remember that presidential elections tend to turn less on the candidates’ position on any single issue than the overall feeling that he (or she) engenders in the public.

“Put more simply: Intangibles matter. And Obama’s polling strength on national security could well help the incumbent win over voters who are on the fence about his overall performance during his first four years in office.”

By drawing a sharp contrast with Romney on foreign policy – arguably the president’s greatest strength – the Obama campaign may be trying to convince swing voters that the president’s overall job performance has been strong enough to merit four more years in office. And if Obama weren’t in a position to cite his greatest national security success so far, that would be a much harder case to make.