why does my integrator perform noise shaping?
The dominant effect you see is from the differentiation which is a high pass function. 

Observe the spectrum after differentiation alone:



As would be clearer on a log log plot the signal is going up at rate f versus frequency. (consistent with a differentiation).

Next observe the spectrum at the moving average output:



This is consistent with expected: The moving average approximates a Sinc filter response, and the nulls of a Sinc function go down at rate f versus frequency. With the differentiator going up at rate f and the moving average going down at rate f we see the flat response as observed with the nulls at $1/T$ where $T$ is the duration of the moving average. 

Your final operation that you call an integrator is actually another average combined with a down-sample (decimation), and with that would result in the low frequency portion of your signal after another 16 point average; so another Sinc filter response of similar form and then the decimated (close in) spectrum out to half of the first main lobe: