Water right sanitizers need to be checked every few years. The KDF media is what kills the iron bacteria and neutralize the chlorine. I have found many dealers do not put the KDF (min-plus) in the softener for several reasons.
1. They find the cost of KDF too expensive
2. They do not understand its use in the system
3. KDF requires high backwash flow rates so it it mis-applied frequently.
If the system was applied for PH adjustment the media is sacrificial and has to be topped off after several years.
The chlorine generator is used to sanitize the media and prevent buildup of iron bacteria on the media. Sanitizers use a form of zeolite media so it is impervious to chlorine. There is also a max amount of H2S that a sanitizer will remove.
If their sanitizer is the old ASC (2510 valve) the chlorine generator may not be working properly especially if they have run out of salt frequently.
Shock chlorinating the well is BIG band aid and as you stated causes more problems than it helps. You keep asking the same questions and we keep replying with the same answers. Do NOT buy another greensand filter or fart around with the present one. Replace it something that works. You have not done any testing or flushing of well which we suggested. Until these things are done and results confirmed all of this is moot.