That's not a controversial equation because it's a simple accounting identity. There's little if any scientific content in it.
This does not follow from the equation. Let GDP = Y. The change in GDP in response to a change in G is equal to
dY/dG = dC/dG + dI/dG + 1 + d(X-M)/dG.
Note that if dC/dG + dI/dG + d(X-M)/dG < -1, then dY/dG<0. In other words, if increasing G causes C + I + (X-M) to decrease, then increasing G might cause GDP to go down. Obviously, this is in all likelihood not going to happen, BUT this conclusion does NOT follow from the accounting identity. You need to know what dC/dG, dI/dG, and d(X-M)/dG are equal to in order to determine what dY/dG is positive or negative or determine the magnitude of dY/dG.
Anyway, most economists do not doubt that dY/dG, the "fiscal multiplier," is positive, but there's a lot of debate about the size of the multiplier, how it changes over time, whether or not it's counter-cyclical, etc. Moreover, some economists believe that although it increases GDP (Y) in the short-run, it might hamper economic growth (dY/dt) in the long-run. These are issues that can be studied empirically or through theoretical models, not simple accounting identities like Y = C + I + G + (X - M). I agree with Opiate that the NYT article misleads when it implies that there is a strong, concrete consensus among macroeconomists about what should be done right now about the US government's deficit.