Because weight is the only practical measure of how much cold mass is actually in the vessel.
The absorbent matrix has to be fully saturated, and saturation takes appreciably longer than most protocols allow. A visual inspection tells you nothing: a partially charged matrix looks identical to a fully charged one. Undercharging is the most common cause of a failed cryogenic transport and it is entirely preventable.
Record both figures, before and after. The difference is your charge. Over time those figures also build a picture of the vessel itself, because a shipper that stops taking its usual charge is telling you something about its condition.
After charging, invert and drain so that no free liquid remains. Free nitrogen changes both the vessel’s regulatory position and its behaviour in transit, and a shipper that sloshes is not travelling as a dry shipper.