Thank you, guys.
I believe, both of two points of view make sense. I should only notice that the qErr is not a white noise (but sometimes is having both periodic and DC components), and also the PI-filter does not eliminate the noise generated by DAC and power supply.
At least, I have not found any significant difference in the output signal of the GPSDO, with and without the qErr correction (ADEV from the GPSDO itself was showing the difference, of course). I made a lot of experiments, receiving QO-100 and telemetry beacons from Es'hail-1 and -2 sats, using my GPSDO for clocking the receiver. I had to pause two months ago, but I hope to continue soon.
As for the built-in TIC, I think we are missing the ability to measure ADEV at 0.01, - or - at least, at 0.1 seconds, since we can only get 1pps as a reliable, lowest-noise output of a GPS receiver. In addition, it is obvious that ADEV plotted from the built-in TIC, does not fully characterize the output signal of the GPSDO.
For the experiments, I built a stand-alone TIC, based on TDC7200 and Arduino M0. This is a home-brew analog of famous PIC TIC and TAPR TICC devices. I will publish all the details in the next message of this topic.
Of course, noise from ADC and power supply should also be studied. It is a good idea to use QO-100 or other sat's beacons to see the quality of the GPSDO's signal just on your screen, simply because thing invisible at 10 MHz become visible at 10 GHz. :-) It looks like the sats are ideal low-cost sources of high-quality 10 GHz signal. As many people successfully use RTL-SDRs and $3-$5 LNBs to receive signals from QO-100, the only task was to synchronize _all_ PLLs to the same 10 MHz output of the GPSDO. I experimented with several types of RTL-SDRs, Pluto and Lime SDR, and also with LNBs taking 24 or 25 MHz as an external clock (it looked like RTL-SDR is sufficient for many experiments).
To convert 10 MHz to a number of other frequencies, such as 28.8, 40, 31.25, 24 and 25 MHz, I developed a simple SI5338-based, Arduino-controlled device with 4 independent outputs. This setup (frequency converter + LNB + RTL-SDR) was very sensitive and was very good to compare several different GPSDOs. Also, it is much more pleasant to look at a live trace on the waterfall of the SDR, than to stare to the still ADEV graph. :-) I will also describe the details in next messages here.