"The pilot must know the center of gravity and gross weight to fly" -- True or false?
True. It is the pilot-in-command's responsibility to determine if the aircraft is safe for flight. 14 CFR § 91.3(a) makes the PIC directly responsible for that. 

The 2016 FAA Weight and Balance Handbook also states:

4. The pilot in command (PIC) has the responsibility
prior to every flight to know the maximum allowable
weight of the aircraft and its CG limits. This allows
the pilot to determine during the preflight inspection
that the aircraft is loaded so that the CG is within the
allowable limits