From what I understand, the softening process puts a ton of sodium into the water
To be accurate, the exchange of sodium with hardness minerals is just that, an exchange. The more hardness minerals - primarily calcium and magnesium - the more sodium is exchanged.
The problem is two fold in terms of brewing:
1. The addition of sodium
2. The removal of calcium & magnesium
The removal of hardness will negatively impact mashing, hop bitterness and so on.
The alternate "salt" used in water softeners is Potassium Chloride which will exchange the potassium ion for the hardness ions. Same problem - different mask!
BTW, for those that say water softeners put salt in your water. Salt is a compound. For what we commonly know as salt for softeners and table use it is Sodium Chloride, one of many types of "salt" that exists as compounds. The ion exchanged in a water softener is only the Sodium ion and it has a metallic taste to most peoples senses and does not taste like "salt" at all. (unpleasantly bitter aftertaste)