The salt in the water promotes rust in steel water pipes. I've literally never lived in a home with steel water pipes, and didn't even consider that. If you have rust, you're water softener has already destroyed your plumbing, and it should be replaced with PVC plastic water pipes, or copper lines if that scares you.
Without the softener, the sulfur smell is quite pronounced (hot showers are unbearably smelly). However, whether the softener is on or off, there is still a significant amount of rust in the water (obvious by how quickly brown-orange rings form in the toilet bowls and the smell which still exists).
We don't have sulfer issues here, mostly calcium in the water. Which is actually good for you, and my tap water tastes very much like bottled water for the same reason. Perhaps a filter system would work better for you?
If the Kinetico people give me a load of nonsense, I think I'll just replace the macro and micro filters (which haven't been replaced in... way too long), bypass the softener, and see how that works out. Maybe it'd even be worth it to go full-retard on the filters and double the number of them (one for large, one medium-large, one medium, then the micro), but filter costs always give me sticker shock.
If your primary concern is the smell of sulfer (it won't actually hurt you, BTW) then a single filter designed for that would be the most cost effective, perhaps an activated charcoal filter?