Consider the following two statements:
(A) Linear momentum of the system remians constant.
(B) Centre of mass of the system remains at rest.
B implies A but A does not imply B.
Consider the (A) statement,
If Linear momentum of the system remains constant then COM moves with the constant velocity. Now you can understand that it need not be zero specifically.
Consider the (B) statement
If COM of the system remains at rest (Here →vCOM is constant and is zero) then obviously Linear momentum of the system remains constant.
Zero is a constant Right!