(note, I'm not the right type of doctor to answer this properly, but my understanding is...)
A lot of chemos are some type of poison, only slightly worse for the tumor than it is for the rest of your body. It doesn't die fast enough (and if it did, you'd be killing the rest of you too). Think months / years with chemo vs. weeks with CAR-T.
I know a child with leukemia who recently started chemo. They had him on sodium bicarbonate and other medication to reduce the acidity of his urine (to compensate for the increased amount of uric acid). However, they only had him on it for the first few days of chemo. My understanding was that the bulk of the chemolysis occured in that period (though I could be mistaken).
It kills specifically. The immune therapies are turning your (or someone else's) body's T cells into murder machines, tuned specifically for a target. And because it is so good at its job, cleaning out the biological waste from destroying cancer is difficult, so trips to the ICU are not uncommon.
Chemo kills everything in your body, just the cancer a little bit more. You do the chemo, wait a few weeks, then do chemo again, wait a few weeks, maybe get a scan, do chemo again, wait a few weeks, ..., maybe in 6 months if your treatments are going well you keep going, maybe they switch you to something else, maybe this round is done and you take a break for a while. You wait because if you don't, the poison kills all of you, not just the cancer.
Why is that not also true in chemo patients?