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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.