The linear momentum is a vector quantity, and the kinetic energy of a particle is always a nonnegative scalar quantity.
Then, the linear momentum of the particles of a system may be such that they balance out when added, while they are not at rest. Then, the kinetic energy of the system can be greater than 0 even while the liear momentum of the system is 0.
Therefore, it does not follow that the kinetic energy of the system is always 0 given that the linear momentum is 0.