We have actually seen the Spudmeister making this mistake:
People often use the GDP formula to erroneously derive conclusions about economic causation. For example, in the wake of the financial crisis of 2008, some proponents of “stimulus” spending argued that a boost in government spending on infrastructure would obviously raise GDP because the textbooks tell us that GDP = C + I + G + (X-M). If the government increases G, according to this argument, GDP obviously must increase, as an increase on the right-hand side of the equation “had to” be balanced by a comparable increase on the left-hand side.
However, the textbook formula does not mean that (say) a $100 billion increase in G must go along with a $100 billion increase in GDP. For all we know, a $100 billion increase in G might cause a $40 billion drop in private consumption (C) and a $60 billion drop in private investment (I). In this case, GDP would remain unaffected, and the private sector would shrink to perfectly offset the growth in government. The textbook GDP formula is consistent with both outcomes, so the accounting tautology, by itself, tells us nothing about the impact of an increase in government expenditures on the economy.