Based purely on the word of the designer/installer, there's carbon in there. And oops, I looked at my notes and I was originally toldto add 1 gallon bleach per month, not week. I don't think I've exceeded that rate. If anything, I did it less than once per month. The weekly instruction came recently, only for me to do for about a month.
Not following the schedule will ruin the filter's media. With carbon, more chlorine by adding it more frequently will also. So now the media needs replacing again.
Attached picture shows this side of the system. Blue tank in back is pressure tank. That feeds into carbon filter in foreground with Fleck controller - cover. That filter is 12" diameter x 50" high. The white tube snaking around that goes to the black water tank. Culligan system is in back right of the picture. I'll also attach a picture of inside water tank. Yellow gunk on bottom is powder, I stirred it up and it re-sedimented after an hour or so.
You have a metered control valve on a filter that requires timely regeneration to prevent the media from fouling, not on gallons used.
Changing that control to a flat cap filter version is not the right way to go, it would be more expensive than buying a 12 day time clock model and selling your "softener" control.
Everything is in in the proper order of installation but you have carbon instead of an oxidizing mineral.
IMO that is wrong because carbon will not remove ferrous/soluble iron or manganese if any of either is present in your raw water. But, those things will use the dissolved oxygen in your water and carbon to convert (oxidize) some of each in the carbon which then loads up the carbon because backwashing can't get the sediment that oxidation causes out of the carbon bed.
So... the filter is basically removing nothing but sediment and H2S gas. The only benefit of the chlorine is bacterial control in the carbon bed and that is highly questionable but will ruin the carbon in a short period of time IMO.
The yellow in the chlorination (salt) tank is probably iron due to the use of raw water in the refill position and the chlorine oxidizing it in the brine tank. As long as the sediment in the brine tank doesn't get deep enough (2"+) to be sucked out during the brine draw position, it doesn't harm anything because that water goes to drain, not into your plumbing. But you can clean out the tank if you want to as long as you know the yellow is simply going to form again.
Your softener looks to be about a 1/2 or 3/4 cuft model. That is awful small.
On both media tanks there should be a label, the model number of the tank will tell you the size of the tanks; I.E. Model 1252 is a 12" x 52" tank which is a 2.0 cuft. IMO you can't get 2.5 cuft or more in that tank.
Your filter tank has a tank adapter in it, the gray part that the control valve is screwed into, and that means your tank has a 4" opening, reduced to the industry standard 2.5" x 8 threads/inch for the control valve to be screwed into. That is a bit abnormal for a 12" dia tank, usually there's a standard tank hole until you get to 14" dia tanks but it's not a problem for you and makes it easier for you to pour the old mineral out when needed. But you don't have 3.0 cuft of media in that tank.