My spin: It’s not so much that a more negative forecast would have prevented Obama from spending large amounts of money, it’s that he would have been forced to tilt the stimulus more in favor of tax cuts which work a lot of faster than government spending (though both are pretty inefficient as “stimulus”).
And Obama wanted to spend billions on his “investment agenda (healthcare, education, infrastructure), not tax cuts. (And if he had spent the $2 trillion that some liberals wanted on stimulus, it surely would have crowded out the rest of his agenda, plus rattled the bond markets.) So he gambled that monetary policy would keep the economy from getting as bad as it has. And he lost.