The press conference Obama held on Friday — at which he claimed the ”private sector is doing fine” — is still being covered as a “gaffe,” and no question, it was a serious misstep. But what also happened on Friday is that a larger argument was set in motion that could be central to the outcome of the presidential campaign.

The two new Web videos the campaigns released this morning — both designed to frame this argument — capture this well. The Obama campaign is out with a new video highlighting Mitt Romney’s Friday claim that we don’t need any more cops, firefighters or teachers. The video broadens the attack to include an indictment of government cutbacks in Massachusetts during Romney’s tenure as governor. “Mitt Romney’s economic plan? He wants to cut jobs for firefighters, police, and teachers,” the video says.

Meanwhile, the Romney camp is out with a new Web video that juxtaposes Obama’s “doing fine” quote with the misery of the ongoing unemployment crisis. “Has there ever been a president who’s so out of touch with the middle class?” the video says.

Obama’s “doing fine” quote is getting more media attention than Romney’s quote about cops and firefighters is. But in a sense, Romney’s quote has at least as much actual policy relevance as Obama’s does. Arguably more. After all, Obama has proposed a plan to address the crisis, which suggests he doesn’t really think the economy is doing fine. By contrast, Romney’s quote is a policy statement — he really is proposing to respond to the crisis by cutting government further.

Obama is pushing a plan to send $35 billion to the states for the hiring of first responders and teachers to speed the recovery. Romney has been claiming that he wants to cut thousands and thousands of government jobs in order to ... put Americans back to work again. That isn’t an actual plan to do anything about the crisis, in the sense that it’s exactly what Romney would be proposing if the economy were doing great.

But as the new Romney video again confirms, the Romney campaign is hoping voters make their decision based on frustration and disillusionment with the pace of the recovery — feelings the Romney camp is trying to exacerbate with the “doing fine” quote — without looking too closely at the true nature of the alternative Romney is offering. The Obama campaign is seizing on Romney’s quote about cops and firefighters to drive home the true nature of the Romney alternative.

Result: The argument over the true relationship between government, public sector jobs and the economic crisis is now front and center. And as it happens, Obama is actually right about the relationship between government job loss and the continuing crisis, and Romney is wrong about it. Surely that’s at least as important as the optics of Obama's admittedly serious gaffe.

* Argument over government is, you know, important: Steve Benen notes that Romney has now positioned himself as the “pro layoffs” candidate, and asks the media to focus on the actual argument between the two men:

The differences between Obama and Romney on this have the potential to drive the presidential campaign: does it help or hurt America when hundreds of thousands of school teachers and first responders lose their jobs?

* The Romney strategy, in a nutshell: Relatedly, Steve Kornacki cuts right to the heart of it:

The Romney strategy is all about convincing voters to throw Obama out while simultaneously selling Romney in the vaguest, broadest terms possible. The idea is to minimize Romney’s exposure in sensitive areas, from culture war politics to specific social safety net cuts, and to position him as a generic, acceptable vehicle for the frustrations of economically anxious swing voters.

* Walker appalled at Romney’s ham-handed demagoguery: The news of the weekend: Scott Walker gently remining Romney that it’s not a good idea to mention cops and firefighters as targets when you’re faulting public workers.

Translation: Dude, if you’re gonna demagogue public workers, do not go after the ones people like! Ever tried “divide and conquer”? Worked for me!

* Yes, government does create jobs: A good column by E.J. Dionne calling on liberals and Dems to stop shying away from that argument and to make it as aggressively as possible. As noted above, the real meaning of Friday’s dueling quotes is that this argument is now joined, if not quite as directly as E.J. would like.

* Obama allies hammer Romney as out of touch: SEIU and the Obama-allied Priorities USA Action are out with a new Spanish-language ad in Florida, Colorado and Nevada highlighting Romney’s silver-foot-in-mouth gaffes to paint him as indifferent to the economic interests of Latinos.

The $4 million campaign reflects a recognition that Romney camp may be able to make inroads with this crucial constituency, despite his positions on immigration, by emphasizing Latinos’ economic suffering.

* Obama campaign to tout auto-bailout in Michigan: The Obama camp is launching a week-long series of campaign events in Michigan to highlight communites in the state that have benefitted from Obama’s decision to bail out the auto industry. The campaign will emphasize the ripple effects the industry’s resurgence has had for other state businesses — an example of government intervention in the economy, which Romney opposed, resulting in an unambiguous success story.

* Union suppport for Obama not monolithic: Gallup finds that a majority of union members support Obama, but nonetheless, 35 percent — or more than a third — back Romney, which Gallup concludes could make a marginal difference in November.

* Did GOP deliberately sabotage the economy? Michael Cohen investigates the case, and concludes:

one of two major political parties in America is engaging in scorched-earth economic policies that are undercutting the economic recovery, possibly on purpose, and is forcing job-killing austerity measures on the states. And they have paid absolutely no political price for doing so. If anything, it won them control of the House in 2010, and has kept win Obama’s approval ratings in the political danger zone. It might even help them get control of the White House.

* And here’s why women support Obama: Shira Toeplitz sat in on a focus group of so-called “Walmart moms” and discovered that they still support Obama because of the strength of the first family’s image and because they are asking whether three years were really enough time to turn things around.

Also: They were surprisingly reluctant to take the risk of changing leadership in the middle of the recovery — an argument that could loom larger than you might think as the campaign proceeds.