Dependence of solubility on pH

I am studying the effect that pH has on solubility of metal salts. One example I came across is the idea that the solubility of $\ce{PbI2}$ is not strongly affected by the addition of strong acid to solution because $\ce{PbI2}$ contains the conjugate base of a strong acid ($\ce{HI}$). Online references explain that this makes sense, since the presence of additional $\ce{H+}$ ions won't successfully pull the $\ce{I-}$ ions out of solution (since $\ce{HI}$) is a strong acid that wants to dissociate).


However, is this line of reasoning actually correct? For example, what if the strong acid added to solution is $\ce{HI}$ itself? Wouldn't that lead to the presence of additional $\ce{I-}$ ions that increase the reaction quotient above $K\_\text{sp}$ and thereby cause the solubility to decrease (more solid to precipitate out)?


I suppose the general question is: how should I think about the role of pH on solubility when the acid being added to tune the pH contains a compound that is being dissolved from the solid in the first place?


