Economics isn't that hard, but David Brooks seems to have trouble with it. He thinks everything would have been different if President Obama had pushed through a $713 billion stimulus that was centered on cutting the payroll tax, had pushed through an infrastructure bank instead of his pork barrel projects, and had focused on getting an energy bill rather than a health care bill.
There is no way of knowing how the politics would have played out, but in terms of the economics, it is difficult to see how Brooks' stimulus would have left us in a different place than the Obama stimulus. With stimulus, size does matter, and Brooks is basically talking about a stimulus of roughly the same size as the one President Obama got through Congress.
Payroll tax cuts are relatively progressive (the tax is regressive), so a cut has a fairly high multiplier since most of the money will be spent. Mark Zandi estimates the multiplier on these cuts at 1.3. This is better than for most tax cuts, but less than the 1.7 multiplier estimated for food stamps, the 1.6 for unemployment insurance benefits or infrastructure spending and the same as the 1.3 estimated for aid to state and local government.
This means that if we compare David Brooks stimulus with President Obama's stimulus of roughly the same size, we should expect it to have roughly the same impact on the economy. President Obama's stimulus included some items that would be expected to have more impact (UI benefits and food stamps), and some that would have about the same impact (aid to the states and his own payroll tax cut, which was called "Make Work Pay"), and some that would have less impact if we include the alternative minimum tax fix as part of the stimulus.
As a result, if President Obama had done the David Brooks stimulus we should expect the unemployment rate to be around 9.5 percent, rising to 9.8 percent this morning, and headed to above 10.0 percent by the end of the year. I can't answer whether this would have made President Obama more popular than he is now.