The "earthing" switch referred to shorts the non-earthed side of the input to the earthed side.
This is commonly used to establish "zero volts" to set the zero line of the display to whatever graticule line is required.
Isn't this only relevant for analog oscilloscopes?
An earthing switch of this nature is trivial to provide in analog 'scopes----just provide another position on the "input coupling" switch.
It gets messier with DSOs ,where the switch is normally a relay,driven ultimately by software.
As I said before,if there isn't an actual earthing switch,a real "hard" earth can be provided by a shorted BNC connector.
If you put a real short across a DSO input,& your display says something other than zero volts ( in other words,an offset),you can see if the 'scope can auto calibrate that out,or just take a note of the offset for later reference.
The use of such is an example of a technique which seems to have been lost in this era of higher stability in test equipment,that of presenting the equipment with a known condition.
People do seem to assume that the accuracy of their equipment is good,& if all goes well that is OK,but in unforseen circumstances,it is good to know whether the DUT has a problem,or your test equipment is lying to you,