Republicans have unveiled drastic budget plans that will either crown their success as radical reformers — or prove a huge misreading of public opinion. President Obama will play no small role in determining which way this plays out.

In the jousting over whether small differences over the 2011 budget will force a government shutdown, Obama has emphasized the importance of compromise, pointedly avoiding the subject of the broad harm in the Republican grand design. In an impromptu Tuesday press conference after talks broke down, Obama said with both pride and petulance that he had already agreed to most of the Republicans’ demands.

But his pride is misplaced. Obama’s eagerness to conciliate only whets the right’s appetite. Consider their plans for next year.

The 2012 budget proposal released Tuesday by Wisconsin Representative Paul Ryan, chair of the House Budget Committee, would cut projected spending over the next decade by $5.8 trillion and further cut taxes. The plan would eliminate Medicare and Medicaid as we know them — and Social Security is next.

Ryan’s proposal to turn Medicare from a government insurance program into a voucher is a stunning gamble. Seniors would get a fixed sum to shop for private insurance. If the money didn’t buy decent coverage, they would have to supplement it with their own resources — or do without.

Medicare is hugely popular. Last September, a Pew/National Journal poll asked about converting Medicare to a voucher. Among respondents 65 and older, just 14 percent supported the idea, while 69 percent opposed it. For all respondents, 33 percent were in favor, while 52 percent were opposed.

Indeed, Obama lost serious political ground when Republicans (inaccurately) characterized Obama’s health reform as weakening Medicare. Now the Republicans are explicitly proposing to dismantle government-operated insurance for the elderly.

The GOP budget would also convert Medicaid, which serves the poor, to a block grant. The federal cost would be capped, leaving states freer to cut already meager benefits. Beyond the poor, about 40 percent of Medicaid outlays finance long-term nursing home care, a benefit that supports the middle-class elderly and spares their families huge expense.

Republicans would slash a wide range of other popular programs from Pell Grants to cancer research. They would drastically reduce funding for public agencies that monitor everything from safe food and drinking water to abusive practices by banks of the kind that crashed the economy.

The Tea Party Republicans seem so besotted with the animated rage of their far-right political base that they are mistaking that narrow energy for a broad shift in public opinion. Yet, in the absence of more clarity and leadership on the Democratic side, they may yet prevail.

Though key Democratic legislators like Senate Majority Leader Harry Reid, and White House press secretary Jay Carney in a little-noticed written release, have decried the extremism of the Republican cuts, the missing figure in this deeper debate is Obama.

On Monday, as Ryan’s budget was leaked, Obama formally announced his reelection campaign in a video and email. The announcement was about strategy, fundraising, and a call for volunteers. The president declined to address this epic national debate about the future of government in protecting the beleaguered middle class.

But what is the next election about? Anything close to the Ryan budget would destroy not only Obama’s own aspirations for America, but repeal core, Democratic-sponsored social insurance anchors dating back to the Great Society and the New Deal.

This Republican-led debate has often seemed like the sound of one hand clapping. You’d think the president would be out there, pointing out that most of the deficit crisis was created by recession and Republican tax cuts, costing $4 trillion over a decade; and emphasizing who gets hurt by these new budget cuts.

Public opinion largely sides with the Democrats’ defense of popular social programs. But that support will remain latent unless a national debate is focused on something that only presidential leadership can achieve. As long as the debate is about who will cut more, the definition of responsible budgeting shifts steadily right, and the Tea Party wins.

Maybe Obama is waiting for just the right moment to draw a bright line. Yet political capital increases most when it is spent, not when it is saved for a rainy day.

The president’s reelection campaign slogan is “Are You In?’’ A better question might be: Is Obama In?