My water softeners have always been in my utility room with the washer & dryer since I built this house in 1977. It is on a standard septic system.
The softener regeneration discharge has always just been a long, smaller size black plastic flex hose just run over and stuck down into the washing machine drain standpipe going into the septic system.
Never had any trouble with any of it. Actually, I think that the softener disharge helps keep the washing machine drain cleaned out with the pressure.
A french drain may eventually plug up.