Plumbing – Is this a workable water softener plan

As a guess, the pipes got layered internally in scale, that will come out over some time (dissolving into the now softer water). Some large chunks may require filter screens, shower heads and valves to be soaked in your vinegar or CLR (containing lactic and gluconic acids).

I know boilers get descaling treatment, but I've not heard of the same with standard pipes. If they were quickly descaled, there might be a large amount of debris, which would necessitate disassembly (washer inlet filters, refrigerator water connections, shower valves, toilet valves). Perhaps all the hose hookup devices could be disconnected for the treatment

Here's my speculation, based on your description and my experience. It is just a theory. Do not commit any resources based on this without collecting corroborating physical evidence supporting it. A direction to investigate at least.

The main floor plumbing goes out to a septic tank. The pipes were not placed lower because there was no need, it just meant digging deeper outside. No one wants to do that without good reason.

When the city sewer was installed, the basement construction tied into it, but the old plumbing was never redirected, it remained on the septic tank. Not the best scheme and possibly illegal, but I'm not surprised someone did this.

It is highly probable there is nothing below to the basement to tie into. The sewer may not be much lower than the basement drainage exit. Anything placed lower than this will likely need to drain into an ejector pumping chamber.