Agreed, why not just use sodium chloride? It couldn't be for health reasons, ie, for a low salt diet, as you are still adding sodium. I find on some of the cooking shows, that while the scientific facts might be correct, the facts espoused are not the ones responsible for the effect being described.
In ion exchange, yes the sodium is readily exchanged for other ions, but in this case probably not Magnesium, but Calcium. I can't think of a big roll for Mg in plants that would account for this effect. Ca on the other hand is a VERY important component in plant cell walls. Remove the Ca, and the walls start to fall apart. But I don't see how simply adding Na would get the Ca to exchange. The Ca - pectin bond is pretty strong. I suspect one would have to alter the pH to break the bond.
In cooking, adding salt will help to draw water out of the cells making it easier for them to collapse and rupture. This would make sense. I can't see where sodium bicarb would do anything different - except raise the pH.
I'd try it with table salt (pH neutral), and then the bicarb(alkali), and then for fun to test the other end of the pH scale, some sodium phosphate (pH 5-7 ish).
I just did a quick literature hunt, and while I couldn't find exactly what I was looking for, I did find the pectins are commonly extracted by treating with soda ash (sodium carbonate - not bicarb - this is more alkali) to create Ca pectate. Ca pectate is not soluble, and its formation will disrupt cell wall integrity. The refs I found were for extracting the pectins for their purification, not about softening cell walls.