..There are many people who make bar soap from wood ash... They add salt but claim to make a bar that is almost as hard as NaOH soap....
Yes, this is a reasonably correct statement. If you use the same blend of fats, just saponify the soap with KOH or NaOH, you will find the all-potassium soap is softer and the all-sodium version will be harder.
The piece of the puzzle that you're missing is that it's not just the alkalis (KOH or NaOH) that affect hardness. The salts also present in the soap will affect the hardness of a soap.
If you add salt (sodium chloride, table salt) to a potassium-based soap at the point when the soap still contains a generous amount of water, some of the sodium from the salt will replace some of the potassium contained in the soap molecules.
That ion replacement will create a mixed sodium and potassium soap. This soap is likely to be somewhat firmer than an all-potassium soap, depending on how much of the potassium has been replaced by sodium.
But you won't be able to get a 100% sodium-based soap out of this treatment. The soap is not going to be as hard as an all-sodium soap made with the same blend of fats.