Why would a Diff Amp require both inputs to have a DC path to GND to work?
That advice is for measuring differential signals. The problem is that, even if the amplifier may have a high common mode rejection (CMRR), the signals have to be referenced to the ground, even if with a high-resistive path.
At the beginning of that section, it says:

It is important to connect the negative lead of a floating source to AI GND
(either directly or through a bias resistor). Otherwise, the source may float
out of the maximum working voltage range of the NI-PGIA and the DAQ
device returns erroneous data.

So if the source is completely floating, the voltage between the ground of the instrument and the input signals may be randomly high. To prevent the PGIA to saturate due to common mode, it suggest to provide a path to ground.
Note
When using a PCB-mounted Diff-Amp (or also an In-Amp), it's not generally required, as the input signals and the Amp will share the same ground.