That statement is BS.
Distilled water (while still not ultra pure which is an expensive process) has (virtually) no ions which is a good thing. Pure water will not conduct electricity which means it will not promote galvanic corrosion between dissimilar metals (iron and aluminum). It will also not leave deposits from the hardness. Additionally it will not have any dissolved elements which will eliminate deposits.
The worst thing you can due is put chlorine in the system. Chlorine is a powerful oxidizing agent.
Soft water is better than hard water, and no the softener does not replace the hard water ions with salt ions, you end up with a small amount of trace salt because salt is used to recharge the resin bed. The resin bed removes the ions. But the softener does remove dissolved elements, only ionized chemical compounds.
Distilled water, with the proper additives is good. The additives promote heat transfer, stop freezing, and inhibit corrosion. i.e. a good antifreeze with distilled water. Since pure water is an excellent solvent, the additives also inhibit any metal becoming dissolved in the coolant.