Why I don't get the right PSD
Since you said you wanted the frequency range to be from 1 $\mu$Hz to 10 Hz, I ran my simulations with 20 Hz sampling frequency and did 100 32k FFT PSDs.  My results looked like what you have. So I lowered the sampling frequency to 0.2 Hz. The sims started at 0 s, ended at 163835 s, with $\Delta$t = 5 s. The WGN was simply unit normal. My results are shown below:



Hope this helps.