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Recent theoretical and experimental work suggest that the honeycombcobaltates, initially proposed as candidate Kitaev quantum magnets, are in factdescribed by a pseudospin-$1/2$ easy-plane spin Hamiltonian with nearestneighbor ferromagnetic (FM) exchange $J_1$ being frustrated byantiferromagnetic third-neighbor exchange $J_3$ and weaker compassanisotropies. Using exact diagonalization and density-matrix renormalizationgroup (DMRG) calculations, we show that this model exhibits FM order at small$J_3/J_1$ and zig-zag (ZZ) order at large $J_3/J_1$, separated by anintermediate phase, which we label as $\widetilde{\mathrm{SL}}$. This$\widetilde{\mathrm{SL}}$ phase is shown to exhibit spin-liquid-likecorrelations in DMRG, although we cannot preclude weak broken symmetries, e.g.weak Ising type N\'eel order, given the limits on our explored system sizes.Using a modified parton mean field theory and variational Monte Carlo onGutzwiller projected wavefunctions, we show that the optimal FM and ZZ ordersas well as the intermediate $\widetilde{\mathrm{SL}}$ state are proximate to a`parent' Dirac spin liquid (SL). This Dirac SL is shown to capture the broadcontinuum in the temperature and magnetic field dependent terahertzspectroscopy of BaCo$_2$(AsO$_4$)$_2$, and the reported low temperaturemetallic thermal conductivity in Na$_2$Co$_2$TeO$_6$ and BaCo$_2$(AsO$_4$)$_2$upon incorporating disorder induced broadening.

Proximate Dirac spin liquid in honeycomb lattice $J_1$-$J_3$ XXZ model: Numerical study and application to cobaltates