I would want to know if military, prisons, and the amount we outspend European economies in health care count as productive output.

They do count, if I understand correctly.

In particular, the figure I quoted has been adjusted to try to account for differences in the cost of living in the 2 countries ("purchasing power parity"), but I doubt the adjustment takes into account the fact that Americans spend more for health care while enduring worse health outcomes than Germans do -- or the fact that American spend more on prisoners while enduring higher rates of crime than Germans do.

(I never meant to imply that the US is superior to Germany, just to illuminate one of the limitations of the OP.)

I get it, and I also suspect that financial services is a large factor is US productivity. And even within financial services there is productivity in, say, retail banking and insurance that most people would equate with productivity in any service function on the one hand, and productivity in creating trillions in derivatives where the risk came to bite us in the arse. Was the "productivity" in that adjusted downward when derivatives fueled the financial crisis bomb? As others have suggested, I'd like to see productivity on a per-industry basis.