Whatching the scope in the last minute of so shows a funny effect: the amplitude is randomly changing quite a lot. I don't think this is a fault of the proble. It is more like a fault of the scope triggering right (a good scope should warn about that - a fluctuating frequency is kind of a warning though hidden). Averaging over shifted traces reduces the signal. It is a kind of trap for the young players when using the averaging mode.
The shields are nice for a tear down, but maybe not too good a contact for a really sensitive signal.
I would be a bit worried about the magnetic core not being laminated very fine. So higher frequencies may not pass through the core as much. They somehow seem to have a good compensation for that, even without the trimmer caps.
Chances are a rectanglular waveform would not look as nice.
For a little more signal one could wrap some 2 or 5 turns through the sensor if needed.
It looks like they just switch the gain of the instrumentation amplifier - so maybe a trimmer for each range.
Averaging 2 back to back sensors could reduce the offset drift - at least the part that correlates between the 2.