> The government sector doesn't produce virtually anything that counts toward GDP…
By a very narrow, technically-correct view, sure.
Preventing, say, people starving to death, dying of preventable disease, unchecked fraud, invasions by other countries, etc. probably helps GDP a bit.
I'd also argue that some services provided by the gov't help towards GDP. An example being the Consumer Financial Protection Bureau and them helping to recoup funds from scams and returning the money to victims who would then spend their money and increase the "Consumption" in GDP.