When President Obama made his famous gaffe last Friday, saying “the private sector is doing fine,” he was trying to make a factual point. Republicans have spent the last three years lashing him as a huge spender and assailing him for harming the economy, but in fact, the current recovery has seen a sharp and unusual contraction of public sector employment. (Ezra Klein has a nifty chart showing this.)

Mitt Romney’s campaign has tried to keep the controversy going. It released its own video purporting to skewer Obama as a hypocrite. The Romney video purports to show Obama first regretting the shrinking public sector, then extolling it:

In fact, as others have noted, Romney’s video is highly misleading. Obama was clearly noting the decline of the public sector as a way of refuting the right-wing caricature of his policies, as the full remarks clearly indicate:

Obama’s gaffe probably hurt his campaign coverage over the weekend, but it may have had the ironic effect of helping him drive home his case.

Why does Romney want to keep this debate going? It’s a way to keep the media talking about Obama’s ungainly line. And it also positions Romney on the side of cutting government bureaucrats, which is popular in the abstract.

But there are also ways in which the debate harms Romney. Seizing on Obama’s gaffe, Romney committed a counter-gaffe, in which he declared of Obama, “He says we need more firemen, more policemen, more teachers.” The flub here is one of excessive honesty. Americans may hate the idea of government in the abstract, but they like it in the specific. The Republican strategy is always to keep its discussion of government programs general — with a handful of exceptions, like foreign aid and programs that help the poor — while Democrats try to make it as specific as possible. Firing police officers, firefighters, and teachers is way less popular than firing government bureaucrats. Obama has taken great care to turn the question into one of those specific job categories, and Romney has inadvertently helped him.

Also, and perhaps more important, the entire controversy has fixed the attention of the news media on the very point that Obama was trying to make: There are many fewer government employees now than there were when Obama took office. Romney is trying to attack Obama for changing his mind on the merits of this fact, but in so doing he is helping to drive home the very existence of this fact.

And it’s a dangerous fact for Republicans to let loose into public circulation. The charge of exploding the government is central to the Republican indictment of Obama. Even if Romney thinks he can win by endorsing the reduction in government employment and charging Obama with trying to reverse it, he is conceding vast ground on the basic terms of debate. The charge of wanting to restore government employment to its 2008 levels is far less potent than the charge that even many swing voters have come to accept. And Obama’s true defense would never have received this level of press attention had Obama not gaffed while trying to explain it.

What’s more, this debate fulfills a second goal of Obama’s: to place himself in opposition to the economic status quo. The broader purpose of his Friday press conference was to remind America that he has an economic plan that Republicans won’t enact. Romney’s general strategy is to force Obama to own everything that has happened to the economy, even those things that have happened over his opposition. Now Romney is endorsing the status quo, and Obama is against it. That is surely the opposite of what Romney wants.