We discuss flavor-violating constraints and consequently possible charged and neutral Higgs bosons phenomenology emerging from a four-zero Yukawa texture embedded within the Type-III 2-Higgs Doublet Model (2HDM-III). We show in detail how we can obtain several kinds of 2HDMs when some parameters in the Yukawa texture are absent. A comprehensive study of the main $B$-physics constraints on such parameters induced by flavor-changing processes is presented, in particular on the off-diagonal terms of such a texture. Having selected the surviving 2HDM-III parameter space, we show that the $H^- c \bar{b}$ coupling can be very large over sizable expanses of it, in fact, a very different situation with respect to 2HDMs with a flavor discrete symmetry (i.e., ${\mathcal{Z}}_2$) and very similar to the case of the Aligned-2HDM (A2HDM) as well as of models with three or more Higgs doublets. We study in detail the ensuing $H^\pm$ phenomenology at the Large Hadron Collider (LHC), chiefly the $c\bar b \to H^+$ production mode and the $H^+\to c\bar b$ decay channel while assuming $\tau^+\nu_\tau$ decays in the former and $t\to bH^+$ production in the latter, showing that significant scope exists in both cases. We also discuss the possible enhancements of the channel decays $H^\pm \to W^\mp V$ ($V=Z, \gamma$) and $\phi^0 \to \gamma \gamma$ that arise at one-loop level.