Understanding what is happening when something is dissovled
Your question seems to be exclusively about salts dissolving to form ions in water. The answer is fairly trivial: a saturated solution of a salt in water is not water.

At (thermodynamic) equilibrium, the solubility depends on the net free energy of the system. When the interactions in solution no longer energetically justify charge separation of the ionic solid into aqueous ions, then you've got equilibrium, right?

So, consider a solution as being (in a very crude and simplified way) Solvent-M1 interactions, Solvent-X1 interactions, solvent-solvent interactions, and M1(aq)-X1(aq) interactions as well as M1(aq)-M1(aq) and X1(aq)-X1(aq) interactions. Which of those will change when you add M2X2 ions to it? Well, at low [M2X2] none of those will change significantly, but you'll add the interactions between M2 and X2 and ALL of the species present. (as already explained, silver nitrate being added to NaCl is a very poor choice, AgCl2 is quite insoluble)

At high enough concentrations you'll begin to significantly change the energetics. In other words, adding M2X2 to a solution of M1X1 changes the interactions of all species present, so you would expect that the mixed salt solution will have different equilibrium concentrations for both M1X1 and M2X2.

If you were to carefully choose highly soluble salts, I think what would be surprising is how well their solubility was predicted by just using each independent Ksp, rather than the (expected) deviations from those predictions. Note that nothing I've said here should be taken to mean that adding a different salt to a saturated solution will decrease the solute concentration of the first salt. (Although it is the most likely scenario, just as the common ion effect would suggest. In some cases solubility will increase!) The reality is that highly concentrated solutions are far more idiosyncratic and generalizations (especially "laws") are much harder to come by.

While there probably is justification to claim that water is dissolved in sodium nitrate (at equilibrium), I've never found such foolishness useful. And I've been around for a long while. (ultra) Pure water does not freeze at 0°C, negative absolute temperatures (negative Kelvin) are attainable, and a bunch of other stuff which while true, doesn't really aid understanding (for the novice) probably should be ignored until one wants to deep dive into a subject.