In this Letter, we propose a simple model that can provide a combined explanation of the $Z\to b\overline{b}$ forward-backward asymmetry, the Cabibbo angle anomaly (CAA), $\tau \to \mu \nu \nu$ and $b\to s{\ell }^{+}{\ell }^{-}$ data. This model is obtained by extending the standard model (SM) by two heavy vectorlike quarks (an $\mathrm{SU}(2{)}_L$ doublet (singlet) with hypercharge $-5/6$ ($-1/3$), two new scalars (a neutral and a singly charged one), and a gauged $L_{\mu }-L_{\tau }$ symmetry. The mixing of the new quarks with the SM ones, after electroweak symmetry breaking, does not only explain $Z\to b\overline{b}$ data, but also generates a lepton flavor universal contribution to $b\to s{\ell }^{+}{\ell }^{-}$ transitions. Together with the lepton flavor universality violating effect, generated by loop-induced $Z^{\prime }$ penguins involving the charged scalar and the heavy quarks, it gives an excellent fit to data ($6.1\sigma$ better than the SM). Furthermore, the charged scalar (neutral vector) gives a necessarily constructive tree-level (loop) effect in $\mu \to e\nu \nu$ ($\tau \to \mu \nu \nu$), which can naturally account for the CAA ($\text{Br}[\tau \to \mu \nu \nu ]/\text{Br}[\tau \to e\nu \nu ]$ and $\text{Br}[\tau \to \mu \nu \nu ]/\text{Br}[\mu \to e\nu \nu ]$).