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
Water softeners do use water to recharge, but it shouldn't be that much. With a new filter and all electronic controls working, most systems should send no more than 25 gallons through during a recharge cycle.
The theory is that the filter media (special polymer beads, often stored in what looks like a high-pressure air tank) is "charged" with a high amount of sodium ions from the rock salt you load into the bin. Hard water containing calcium and magnesium carbonate, which are only slightly soluble and cause lime scale, comes into the filter from the house supply, and the sodium replaces the calcium, forming highly water-soluble sodium carbonate in the water supply, which won't cause scaling, leaving the calcium and magnesium ions in the filter. To "recharge", the softener fills the salt bin with supply water producing a brine, which is then flushed through the filter media. The sodium replaces the calcium and magnesium in the filter, which then hitch a ride down the drain as soluble chlorides.
The amount of water needed to produce the recharging brine varies by manufacturer and age of the filter media; this ion exchange does take its toll, as the charged ions are corrosive (a surplus of positive ions in a water solution makes the solution acidic). The softener also relies on several detectors to determine system status, that are exposed to hundreds of thousands of gallons of water over their rated life, much of it salty and highly corrosive. Over time, all these components wear out, with more or less detrimental effects on the system's overall efficiency.
Best Answer
The main water line will go in the house far enough to get a good spot to tie in a water conditioner. I had mine in the crawlspace, it was 3' tall and I dug it into the dirt to get the height needed to fill and maintain it, also to protect it from freezing (Maryland at the time). Since I did not want it in the living space taking up room, that is why the crawlspace. Most conditioners are placed in the living space perhaps the laundry room, or at least the garage, depending where the water main is accessible. Any competent installer can run the pipe where it is needed to install the system.
My suggestion is to get or try another installer. Always good to get more than one estimate. My conditioner was for the whole house, not for just the cold or hot. I did not have the outside hose bibs tied to it, no need for that.