Motivated by recent experiments [P. Vaidya {\em et al.}, Science {\bf 368}, 160 (2020)] on spin pumping from sub-THz-microwave-driven uniaxial antiferromagnetic insulator (AFI) MnF$_2$ into heavy metal (HM) Pt with strong spin-orbit (SO) coupling, we compute and compare pumped spin currents in Cu/MnF$_2$/Cu and Pt/MnF$_2$/Cu heterostructures. Recent theories of spin pumping by AFI have relied on simplistic Hamiltonians (such as tight-binding) and the scattering approach to quantum transport yielding the so-called interfacial spin mixing conductance (SMC), but the concept of SMC is inapplicable when SO coupling is present directly at the interface. In contrast, we use a more general first-principles quantum transport approach which combines noncollinear density functional theory Hamiltonian with Floquet-nonequilibrium Green functions in order to take into account {\em SO-proximitized} AFI due to adjacent HM; SO coupling at interfaces; and evanescent wavefunctions penetrating from Pt or Cu into AFI layer to make its interfacial region {\em conducting}. The DC component of pumped spin current $I_\mathrm{DC}^{S_z}$ vs. precession cone angle $\theta_{\vb*{l}}$ of the N\'{e}el vector $\vb*{l}$ of AFI {\em does not} follow putative $I^{S_z}_\mathrm{DC} \propto \sin^2 \theta_{\vb*{l}}$, except for very small angles $\theta_{\vb*{l}} \lesssim 10^\circ$ for which we define an {\em effective} SMC from the prefactor and find that it doubles from MnF$_2$/Cu to MnF$_2$/Pt interface. In addition, the angular dependence $I^{S_z}_\mathrm{DC}(\theta_{\vb*{l}})$ differs for opposite directions of precession of the N\'{e}el vector, thereby leading to twice as large SMC for the right-handed as for the left-handed chirality of the precession modes of localized magnetic moments within AFI at resonance.