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I agree with @jks. The dominant source of error is ionospheric propagation. In addition, not all KiwiSDRs report their position accurately, i.e., their positions are either put manually by the owners or get rounded in order not to reveal the exact location of a given KiwiSDR.
Also consider the fact that one sample at 12000 samples/sec corresponds to 25 km, and the bandwidth of most signals on shortwave is less than 12 kHz. So if the bandwidth of a signal is 3 kHz the corresponding pseudo-range resolution is 100 km.
Having said that, it would be interesting to compare the stability of the KiwiSDR GNSS-derived timestamps w.r.t. ones from a GNSS-disciplined rubidium standard.
To null, use SAS mode and select the "audio" tab on the main control panel. On the bottom SAM line you should find the "channel null" menu. Select "null LSB" or "null USB". Make sure the PLL carrier is around 0 Hz (+/- 100 Hz maybe). If it's way off then make sure you're tuned to the station center frequency, switch to AM mode briefly, then back to SAM. The PLL should lock.
Did you mean SAM mode? It's the only mode in which I find the "channel null" menu.