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I would rather buy a vacuum tool that removes all the air from the diff rather than some checker. Regular rebuilds cost less than one checker. Having several diffs built at different CTS, I personally see this as a vanity tool while it may provide consistency in theory with numbers, the best method is to rebuild as the data is presented in variables that often times are irrelevant. My point imho is that if you open a diff to top it off.. change the oil, if it is leaking replace bad seals, outdrive, and or, case I believe this will provide more consistency hard or soft. After all the engineers that developed the oil are in theory correct.

Removing the air bubbles from a diff is pointless. You will trap air in it when you seal the case, and the moment you spin it up, it emulsifies the air back into the oil.

The experience as with shocks is that more or less a drop of oil can make a huge difference. Going a small step lighter in oil but using a drop of extra oil can make a diff and a shock harder. So a meter will check if you did it right. In case of good drivers I think it is wise to see if your change is established or was is wrongly done.
It is like you want more toe-in and screwed it up with a toe out because you did not measure it..

Great work! I had no idea I needed a new tool/project until now!
Stupid question: How (and when) do you use this if the scale is in current and not a torque value? Do you just check the difference between front and rear diffs?

You can check stiffness of different diffs. The valor is not important, the important is the difference between 2 numbers (ex: 1600 is stiffer than 1400)