When President Obama invited Michael McCaul, R-Texas, chairman of the House Committee on Homeland Security, to visit the White House, he got an unexpected response. McCaul said no.

"I saw it as a political trap," McCaul told conservative radio talk-show host Laura Ingraham.

Since Mr. Obama first entered the Oval Office, he has never presented a serious strategy for pushing his immigration amnesty bill through Congress. Instead, the White House has just done enough to keep the hopes of amnesty advocates alive—softening enforcement of immigration laws, killing cooperative programs with state and local law enforcement, granting waivers to delay deportation orders, etc.

But if the White House were to decide it really should launch a full-court press for amnesty, it would find that job is now a much heavier lift, due to the continuing debacle of the Obamacare rollout.

Over time, the president’s massively complex overhaul of national healthcare has become more, not less, controversial. And it never was popular to begin with. Having Mr. Obama take the lead on selling a similar “comprehensive” approach to immigration and border security reform might do more harm than good to the cause of amnesty advocates.

Thus far, the Oval Office has consigned itself largely to the sidelines, opted simply to cheerlead for the Senate’s “Gang of Eight” bill. That bill passed the upper chamber in June, and has gone nowhere in the House.

As for getting that bill to the president’s desk for signature, the strategy now seems to be strictly “hope and change”: The White House hopes the House will pass some small-bore immigration bill, which it can then change into an amnesty bill in conference.

Rep. McCaul’s calling out of Mr. Obama is the latest signal that House leaders don't want to play that game.

Indeed, House leaders seem to have embraced a game-changer in the immigration debate, a way to advance a positive agenda without playing into the president's all-or-nothing amnesty option.

The reality of the immigration debate is that all responsible sides in the debate want America to remain a nation of immigrants. They want secure borders that let the legitimate movement of goods, people and services flow, while stopping the illicit traffic in people, guns, drugs and money. What the House has always lacked is a clear strategy for moving toward those common goals without falling into Mr. Obama's amnesty trap.

McCaul has been one of the leading advocates of this alternative strategy. He helped make the case with leadership that the right answer was a step-by-step, piece-by-piece approach that starts with initiatives we can all agree rather than the troubling issues that divide us.

If the House continues its strong stand, a number of good things might happen. The most important might be killing the Congressional appetite for complex, unmanageable bills like Obamacare. As is becoming increasingly clear, “comprehensive” legislation creates as many problems as it purports to solve, creating a desperate stampede for waivers from the thousands of pages of regulations subsequently promulgated to try to implement the bill.

Another benefit might be actually making progress on immigration and border security reform, hopeful starting with lawful immigration and nonimmigrant visa programs. Small steps, like expanding the Visa Waiver Program, will increase jobs and economic activity at little cost and without rancor.

Congress might also concentrate on taking its oversight responsibilities seriously. The executive branch needs to be held accountable for upholding the rule of law. Our nation needs sensible, cost-effective border security and diligent enforcement of existing immigration laws. Admittedly, neither political party would benefit from this approach—but it would start to solve the challenge of broken borders and a flawed immigration system.

The White House might well decide it doesn't want to play the House's game, rejecting a step-by-step approach and continuing its all-or-nothing quest for amnesty. Neither political party really benefits from this either.

But stalemate gets us no further toward mending our broken immigration system. That scenario leaves all Americans as losers.

- James Jay Carafano is vice president for foreign and defense policy studies at The Heritage Foundation.