I'll stay out of the fray of pH and TA, you're getting good advice there. Just some observations...
Add your softener hook up to your signature, something like mine.
You likely are not auto-filling with zero-CH water. I thought for sure I was, but my CH crept up over the last few years to the point where I had to do a water exchange to bring my CH back down. Of course, I made the mistake of doing so last fall, and then we got all the rain, but it's still around 325, so I'm good.
The CH rise is a bit of a mystery, but I've got a strong theory. I don't think it's leaching out of my plaster, because my CH started at about 350 and got up to over 500. At some point there would have been plenty of CH in the water and any leeching would have stopped. So from where then?
I, too, can test my softened fill water and get zero CH. But it's probably not zero, it's just not registering. So with our very active evaporation, we're putting upwards of 100 gallons of water into the pool a day, and so some amount of CH each day. As you know, CH won't evaporate out, so it's going to collect. And on your craziest swim days, I can't imagine your splashing 100 gallons of water out of your pool each day, so you are collecting CH, even from softened water. And even with crazy splash out, you'd be refilling with the same low-CH water, so you would not lose much CH, you'd just be breaking even. Except...
The other likely source of CH is from the unsoftened water you're adding some number of days each week. Between the pool and your family, your water softener is probably regenerating somewhat often. (Do you know how often?) Every time it does, it goes into bypass mode for a few hours while it does its regen thing. Well, when your auto-filler kicks on while your softener is in bypass, it'll be drawing water from the street, with all the CH and TA goodness your city provides. And that CH will collect, too.
Point being, you're very likely not auto-filling with CH-free water. Do what you're going to do with pH and TA, and if you then determine you want to add a little CH, add only the bare minimum to make your CSI happy, because you shouldn't assume it's not going to rise, it likely will. If CA continues to have decent winters, your CH will probably stay stable, as mine did for the first few years, because each winter you'll get a free water exchange. But if (when) we go back into drought mode, you won't get enough exchange, and so your CH will rise and you'll eventually need a manual exchange. Mine took several years to get too high, but it did.
I've addressed my softener bypass issue, and will eventually explain how I did it in a thread I've yet to write. I'll keep it updated in the coming years to track my solution, which might prove my theory. Short version, I use a motorized valve and my pool automation controller to shut down the water source to my auto-filler during the hours my softener regenerates, such that my auto-filler, even if it activates during those hours, won't be pumping any hard water into my pool. So now the only source of CH my pool will be getting is whatever sneaks past my softener, which I'm hoping will be offset by my free winter-rain exchanges.