It would be helpful to know whether you are on a water system or a well and what are the specific conditions of your water. Also, what brand and model of softener you have and how many people are in the home.
"This is my second water softener, and am convinced (from an engineering standpoint), that a robust water softener has yet to be designed
That will come as news to the million upon millions upon millions of water softener owners worldwide that enjoy soft water for decades with a minimum of maintenance.
Here are tips that might help you... never put in more salt in the brine tank than necessary to cover the water. The more salt in the brine tank the more weight on the salt at the bottom and the quicker it will mush or bridge.
Once a month, take a gallon of hot tap water and slowly pour it down the brine well (the round 4" tube in the brine tank where the float lives). That will help dissolve any salt that has "bricked up" at the bottom of the brine tank and around the brine pickup.
Once a year take your brine tank outside (if it has a seperate brine tank) and remove all the salt and thoroughly clean the inside. Don't start chipping away salt with a screwdriver cause you will damage the brine tank
. Any hardened salt can be dissolved with hot tap water. If your softener is in the garage and the water heater is there a hose from the water heater drain works perfect to dissolve hardened salt and that's a good time to drain the water heater, done that yearly ? If your softener is an all-in-one cabinet design then you have more work to do and may have to clean out the brine area where it sits. When you're done, add a couple gallons of water to the brine tank and a bag of salt. After the softener regenerates add salt to cover the standing water and no more.
A correctly sized and setup softener will regenerate every 6 or 7 days, more often if there is an iron problem. A longer interval between regenerations can cause salt problems and be hard on the resin.