Why is it impossible to write a net ionic equation for any mixtures that did not form a precipitate?

The net ionic equation shows the ions that have participated in the reaction. All other ions are spectator ions. If an ion starts out aqueous, and remains aqueous, it hasn't done anything. It is a spectator. Actually a net ionic equation can show the formation of a precipitate, a gas, or water. Reactions where these products form are end reactions. Reactions where none of these products form are reversible and are simply a mixture of spectator ions.