  Recently, honeycomb cobaltates with 3$d^7$ were proposed to display Kitaev
physics despite weak spin-orbit coupling. However, other theoretical and
experimental works found leading XXZ Heisenberg and negligible Kitaev
interactions in BaCo$_2$(AsO$_4$)$_2$ (BCAO), which calls for a further study
to clarify the origin of the discrepancies. Here we derive the analytical
expressions of the spin model using strong-coupling perturbation theory. With
tight binding parameters obtained by {\it ab-initio} calculations for idealized
honeycomb BCAO, we find that the largest intraorbital $t_{2g}-t_{2g}$ exchange
path, which was assumed to be small in the earlier theory proposal, leads to a
ferromagnetic (FM) Heisenberg interaction. This becomes the dominant
interaction, as other $t_{2g}-e_g$ and $e_g-e_g$ contributions almost cancel
each other. Exactly the same assumed-to-be-small channel also generates an
antiferromagnetic Kitaev interaction, which then cancels a FM Kitaev
interaction from $t_{2g}-e_g$ paths, resulting in a small Kitaev interaction.
Under the trigonal distortion, the preeminent isotropic Heisenberg becomes an
anisotropic XXZ model as expected, which is the case for BCAO. However, in
Na$_3$Co$_2$SbO$_6$ the intraorbital $t_{2g}-t_{2g}$ hopping is smaller and
comparable to the $t_{2g}-e_g$ hopping, leading to a delicate competition
between enhanced Kitaev and reduced Heisenberg interactions.
