No offense to your grandmother, but she doesn't sound like a chemist either. Fats go rancid because of heat, light, and oxygen. Properly storing in a freezer removes all three of these factors.

If anything, salt will make fats stay fresh longer - for instance, most generations prior to widespread refrigeration technology grew up on salted butter. Fresh butter was a special treat, only available during certain months. Salted butter was the norm, because the salt was used as a preservative, and would allow the butter to be kept at cellar temps for months and months, all the way through winter and into spring. Yes, the result was more salty (they used 10x the amount of salt we use in salted butter now), but the butter fat didn't get the flavor of rancid fat (which, seriously, is so absolutely fucking vile that once you have it pointed out to you, you'll never not notice it - no amount of salt, seasoning, or herbs can cover up the flavor of rancid fat).

If you're freezing to 0, you're fine, regardless of salt content.

Also, for the love of god, salt your food while cooking! The only cultural reason we add salt afterwards is because a French king hundreds of years ago did, or because we're cooking for someone with strict sodium limits.