When speaking of "mechanically softened but using salt, I might guess that it is more correct to call it an ion exchange softener. It is done mechanically but the process is to replace calcium and MG ions with sodium ions and that is where the salt is used. You are correct that most of the salt is not left in the water but used as a brine rinse for the softener media and then flushed down the drain. You need to keep replacing the salt because the salt keep getting flushed down the drain. However there is obviously SOME salt left after the rinse, just as there is some soap left on your plate after washing and rinsing. The amount of salt the normal person takes in from drinking softened water is often compared to what one finds in a single slice of white bread. So if you are on a strict diet to avoid salt, potassium is suggested as the better way to soften the water.
But the salt in soft water is not the problem as I see it. I use softened water to feed the irrigation on one side of my house in Texas and it doesn't bother the grass at all. There are plants which thrive in salty environments and there are those who will not. The air within a couple miles of the coast is often more salty than softened water. Think Florida?
I find the softened water is lacking in all the minerals that I need so I have a choice of using it and fighting to add those back or finding a way to tap into the line before it goes to the softener. Adding a tap was far easier for me. In Texas may softener is just through the wall from the fish room so punching a whole was ideal.
If you are anywhere in the West County area, you definitely will need the softener but , maybe looking at how it is plumbed would show an easy way to tap in before the softener?
At the top of the softener, there is likely to be a bypass with possibly arrows showing which is going in and out.