A seminal time for his presidency: a true test of his leadership. Little wonder, you might think, that Barack Obama's approval ratings have soared in the wake of the targeted killing of Osama Bin Laden – from the lowest of his presidency to the highest in years. A series of polls shows that some 56 per cent of Americans now approve of his performance in his job as president, while more than two-thirds aprove his handling of the terrorism threat.

It's not surprising, when you read the accounts of Obama's coolness under pressure as he watched the live feed of that daring raid by Navy SEALs . . . when you consider that while making one of the most important decisions of his career, he managed to pull off a creditable comedy riff to a crowd of hacks at the White House correspondents' dinner . . . when you bask in the brief euphoria of what the pundits are calling the "USA! USA!" moment the country hasn't experienced for years.

It's ripped up another stereotype, too: that of Obama as hand-wringer in chief, an image of indecisiveness and weakness that has dogged him almost from the moment he took office. Suddenly his harshest critics – yes, even the likes of former Vice-President Dick Cheney and Obama's talk-show nemesis Rush Limbaugh – were lining up to offer praise. As the FT put it, "Mr Obama has a compelling new narrative."

So, it's not surprising, either, that the president tried to sieze the opportunity to build some of that political unity everyone's been paying lip-service to since, well, for ever. In his speech to the American people – which 56 million people tuned in to watch – he was pretty optimistic. As he said: "It is my fervent hope that we can harness some of that unity and some of that pride to confront the challenges we still face."

Yeah, right. It lasted all of about 12 hours before the Republicans went back on the offensive, with the Democrats hard on their heels. Here's the Republican senator Jim DeMint, not long after he had offered his "heartfelt thanks" to the president for "pursuing the neccessary policies to bring about today's success", changing his tune rather radically to weigh in to the adminstration's economic record. "The command-and-control paranoia that we see in this administration is antithetical to everything that we understand about freedom in our country," the senator opined.

And Tom Pawlenty, who's off to South Carolina to debate his GOP presidential rivals on Thursday said that "the time to engage President Obama is now" – and that despite the Bin Laden killing, Obama's "national defence posture" won't be off limits.

Nor did politics on the Hill take long to get back to the usual partisan sniping, with Republicans railing against the White House about everything from petrol prices to light bulbs (yes, really). The Democrats have hardly been sitting on their hands, either – barely pausing for breath before scheduling a press conference attacking the Republican plans for Medicare.

So the glow of national pride of the past few days, that "USA!" moment that echoed around Ground Zero, could very well turn out to be just that – a moment – as the normal dynamic of the 2012 campaign takes over once again. And despite the events of this week, that dynamic is likely to be driven by one thing: the economy. Take note: Obama's ratings here are by no means as healthy as his handling of the terror threat, and are still languishing at an all-time low.

There's no doubt that the president has proved his credentials as commander-in-chief: it'll be hard to keep asking that "3am phone call" question. But by the time Americans go to the polls next November, it could well be jobs, gas prices, the debt and the deficit that decide the way they vote. Without doubt, the economy will be the GOP's biggest electoral opportunity. And, blink and you'll miss it, but Obama and his campaign team could just have their best chance yet to make sure it's no longer their biggest electoral liability.

Felicity Spector is a deputy programme editor for Channel 4 News.