  The Hubbard model on the kagome lattice is presently often considered as a
minimal model to describe the rich low-temperature behavior of AV$_{3}$Sb$_{5}$
compounds (with A=K, Rb, Cs), including charge-density waves (CDWs),
superconductivity, and possibly broken time-reversal symmetry. Here, we
investigate, via variational Jastrow-Slater wave functions, the properties of
its ground state when both onsite $U$ and nearest-neighbor $V$ Coulomb
repulsions are considered at the van Hove filling. Our calculations reveal the
presence of different interaction-driven CDWs and, contrary to previous
renormalization-group studies, the absence of ferromagnetism and charge- or
spin-bond order. No signatures of chiral phases are detected. Remarkably, the
CDWs triggered by the nearest-neighbor repulsion possess charge
disproportionations that are not compatible with the ones observed in
AV$_{3}$Sb$_{5}$. As an alternative mechanism to stabilize charge-bond order,
we consider the electron-phonon interaction, modeled by coupling the hopping
amplitudes to quantum phonons, as in the Su-Schrieffer-Heeger model. Our
results show the instability towards a tri-hexagonal distortion with $2\times
2$ periodicity, in a closer agreement with experimental findings.
