But some variant of the New Deal approach would be possible. For instance, instead of paying people not to work for 99 weeks (as we currently do), we could pay people wages equal to unemployment benefits for jobs that would otherwise be done by a unionized government employee. For instance, unemployed skilled workers could become DMV clerks and unemployed unskilled workers could become DMV janitors.
Such an approach would allow us to both create jobs and cut spending. Strangely, we don't go down that route. It's almost as if our elected officials and unelected bureaucrats aren't looking out for our best interests.
For megaprojects where a municipality might be creating new infrastructure or modifying infrastructure, sure. But there are plenty of big projects involving deferred maintenance that would require little or environmental impact work (or may have even had the environmental work done long ago).
Of course, this assumes that producing services is the government's actual goal. If funneling money to cronies (politically connected corporations, government unions, etc) is the goal, then this method won't accomplish that one so easily.
That is indeed a way that the society you live in choose to spend money to create jobs. Other ways of spending money to create jobs are, for example to increase the size of the military, which may or may not be as useful for a nations economy as creating infrastructure, depending on the state of your infrastructure and the size of your military.