You don't know much about the New Deal then. First the down turn in the economy ended mid 1933. The rebound had already started before FDR had passed a single bill or bit of legislation.
FDR actually had two "New Deals". The first one which he past on entering office, cut taxes, reduced federal spending, and reduced government work force, and slashed government salaries. In other words, he acted like a conservative. Under this the economy did well.
However, by 1935 to 1937, he started massive government spending. The national debt grow, the economy slowed, and eventually a recession hit in 1937.
Government contracts are all temporary, and companies know this. Once the work is completed, without real economic increase, all the people will be instantly laid off again, and everything will fall back to prior levels.
Actually we don't "need" schools. For the past 8 years, my city that I live in, has wanted to build a second high school. Each year they try and pass a new massive tax levy to build it. The economy has been good, the economy has been bad, without any new school. Are you suggesting that if we build the school, the economy will suddenly boost? No of course not.
As for roads, we have them. Are you suggesting that if Obama blows a few billion on roads, that our economy will return? Of course not.
Blowing several billion, to save a few thousand in fuel costs, or a few hundred in maintenance, is not going to help. It will require higher federal taxes, that will offset any small savings.
Not sure how YOU mean that, but yes War is the perfect example. After WW2 was over, and the artificial demand created by government was ended, the country slumped back into a recession. Why? Because the real economic level was still much lower than what was artificially created by government.
Similarly, Obama's government works project will only have an artificially positive effect for as long as it exists.
That's the dirty secret about economic activity. Politicians can manipulate economic numbers simply by blowing money. This is because GDP includes all government expenses. Want to "make the economy rebound" instantly? Blow tons of cash.