You left off a key phrase from the paper. Noncentrosymmetric systems with time reversal symmetry don't have nonreciprocal linear response. Linear response on its own doesn't have this property. For example Electric Polarizability is the linear response of the dipole to a change in the electric field and this in general is nonzero for noncentrosymmetric and centrosymmetric systems. Nonreciprocal linear response seems to look at the difference in linear response for perturbations with opposite helicity/chirality/handedness. Reversing time also reverses the direction and helicity of the wavevector, so time reversal symmetry is related to this. So I would think the nonreciprocal polarizability would be zero, since using left vs right circularly polarized light should make no difference.
As to why nonreciprocal linear response is forbidden (at least, as mentioned in the paper, for the diagonal elements). This comes from these Onsager reciprocal relations, which relates some linear response function to a similar, time reversed response function: $$\tag{1}K_{AB}(q)=\epsilon_A\epsilon_BK_{BA}(-q)$$
This just says that the response function for operators A and B (A input, B output) is equal to the response function for B and A (B input, A output) with the wavevector flipped and the sign flipped based on the time symmetry of A and B. For a diagonal element (say xx), reversing the order of A and B doesn't matter (it's still the xx element of the same tensor) and assuming time reversal symmetry for each operator, this reduces to $$\tag{2}K_{A_xB_x}(q)-K_{A_xB_x}(-q)=0$$
This is exactly the nonreciprocal response for this element, so we can see that it will always vanish for a diagonal element. For an off-diagonal (say xy), reversing the order of A and B does matter (the reciprocal theorem relates the xy and time reversed yx elements of the tensor) $$\tag{3}K_{A_xB_y}(q)-K_{B_yA_x}(-q)=0$$ For the nonreciprocal response to be zero, the second term would be $K_{B_xA_y}(-q)$. The relation doesn't say anything about the nonreciprocal linear response in this case, so in general these elements will be nonzero.