Below the solubility limit of a salt there is no trace of the original ionic solid. Such a solution is homogeneous. However above the solubility limit undissociated salt coexists in equilibrium with the electrolyte. Such a saturated solution is heterogeneous.
Electrolyte solutions are in dynamic equilibrium. Hydrolysis reactions that convert sodium ions and water into $\ce{NaOH +H^+}$ or chloride ions and water into $\ce{HCl +OH-}$ might happen, but the products $\ce{NaOH}$ and $\ce{HCl}$ then quickly dissociate and hydroxyl ($\ce{OH-}$) and hydronium ($\ce{H+}$) quickly recombine to form water again*.
$\ce{NaOH}$ and $\ce{HCl}$, if present, are solvated species, which means they are surrounded by water molecules that interact non-covalently. Interactions with the solvent are indicated by writing (for instance) $\ce{HCl(aq)}$, where "aq" is shorthand for aqueous (solvated by water). But again, such species are highly unstable in water and will rapidly dissociate into ions, mainly because oxygen and chlorine are highly electronegative and readily remove an electron from sodium and hydrogen respectively, and the separated ions are strongly stabilized by interactions with polar water molecules which form a solvation sphere (or hydration shell).
Water itself dissociates into hydronium and hydroxyl ions (by reacting with other water molecules) but the equilibrium again strongly favors formation of water rather than the persistence of the ions above a limiting ion concentration (or rather product of concentrations, as determined by the dissociation constant of water). Above the limit the reverse reaction between available $\ce{OH-}$ and $\ce{H+}$ occurs with a very high likelihood and therefore faster than the forward self-dissociation.
*Sodium ion might be seen here acting as a catalyst in the autodissociation of water, as indicated by writing $$\ce{H2O + Na+ -> NaOH +H^+ -> Na+ + OH- +H^+}$$, but these are speculations on my part. However catalysis of water autodissociation, for instance in bipolar membranes, is an active area of study. And given the importance of charge separation during autodissociation, the possible role of stabilizing charges would not be surprising.