Some people dislike even a hint of chlorine smell or taste, and other people are convinced chlorine has, well, long term adverse health consequences...
An alternate method to neutralize formed biofilm (slime) and sanitize the house water system would be the use of hydrogen peroxide.
It is a strong oxidizer at the strength sold in stores, so it would need to be diluted.
Most house water systems today are all plastics, including the part of the pump that the water passes through. Older pumps might have metal inside the water housing/pump, so use with care/caution if you have an older pump. Today's plastic lines/hoses, pumps, valves, tanks, etc, would be (should be) fine with dilute hydrogen peroxide.
The advantage of hydrogen peroxide is that over time it breaks/deteriorates into the more stable/common compound we know as WATER..
Hydrogen peroxide is less aggressive as a sterilizer/sanitizer than chlorine, so significantly more than a couple capfulls per tank would be needed to do the job.
If you have a big house tank, and a newer all plastic system, maybe add the whole quart (it is already quite dilute when you buy it in the store) to the full tank, run it into the lines to sit for an hour or so, then flush with fresh water. With smaller tanks adjust down amount added accordingly. I would not let it sit in the system long term because of an abundance of caution.
Extremely dilute, it has no flavor, so rinsing can be once and done.
I have not done this with my system, I use chlorine, so, consider the source. I KNOW it will kill/oxidize the biofilm (slime), and modern plastics are resistant to dilute solutions. The amount suggested is approximate, but a little strong should not cause problems in an all plastic system.
I'm not trying to strong arm anybody. This is presenting an alternative to a point of resistance with some people.
Do what you are comfortable with after you have thoughtfully considered the information in this whole thread.