What would cause a brand new water softener to not soften water?
Turns out @Ecnerwal was correct about my testing being flawed.  The softener appears to be working fine.  As he mentioned, it makes 100% sense that a TDS meter will measure exactly the same before and after a water softener install since the water isn't being filtered.

I'm not sure if my initial user experience tests were simply done too soon or if it takes a couple days to clear out the pipes but the slippery feeling of the water is now present as expected.  I have also been able to verify that everything is fine with the softener by the fact that I can run the dishwasher without rinse aids for the first time in over a year.



In this picture, the cup on the left just came from the dishwasher.  The cup on the right came from my cupboard and was ran through using a rinse aid before the softener was replaced.  Previously without the rinse aid, you wouldn't have been able to see my fingers through the cup.