  Recent theoretical and experimental work suggest that the honeycomb
cobaltates, initially proposed as candidate Kitaev quantum magnets, are in fact
described by a pseudospin-$1/2$ easy-plane spin Hamiltonian with nearest
neighbor ferromagnetic (FM) exchange $J_1$ being frustrated by
antiferromagnetic third-neighbor exchange $J_3$ and weaker compass
anisotropies. Using exact diagonalization and density-matrix renormalization
group (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 an
intermediate phase, which we label as $\widetilde{\mathrm{SL}}$. This
$\widetilde{\mathrm{SL}}$ phase is shown to exhibit spin-liquid-like
correlations 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 on
Gutzwiller projected wavefunctions, we show that the optimal FM and ZZ orders
as 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 broad
continuum in the temperature and magnetic field dependent terahertz
spectroscopy of BaCo$_2$(AsO$_4$)$_2$, and the reported low temperature
metallic thermal conductivity in Na$_2$Co$_2$TeO$_6$ and BaCo$_2$(AsO$_4$)$_2$
upon incorporating disorder induced broadening.
