Your “wrinkle” is correct. And that is how fiscal policy is normally presented in NK models. But it is not the increase in G(t) that increases AD (as is commonly assumed), it is the decrease in Gdot(t) that immediately follows that increases AD. And realistically, G(t) is not a jump variable in the real world. Governments cannot suddenly decide at time t to jump G(t) up (or down). There aren’t enough shovel-ready cliches. It takes time to move G(t), so in practice Gdot(t) is the only control variable.

I did a simpler clearer version here: http://worthwhile.typepad.com/worthwhile_canadian_initi/2013/11/on-understanding-and-spinning-nk-models.html