I think the fact they rapidly multiply while killing cells, inside your body, without any form of control beyond the initial "programming" deserves a, huge, honking, warning sign somewhere in that sentence. Once they go in, they may cure cancer, they may kill you in less than an hour, they can destroy your immune system and they can have your immune system destroy you.
Just to make clear this treatment will always be in the "you'll survive next week? This treatment is too risky" class.
Don't get too excited, this is medicine right? Surgeons literally cut people open -- don't try this at home.
CAR-T these days is indicated after hugely toxic therapies like BMT have failed. Plus they understand the danger of the cytokine storm responses and are at the ready to mitigate. A successful CAR-T therapy is massively less toxic overall.
If you really want to get freaked out: CAR-T uses a modified, neutered virus to reprogram the T cells. Today that's some lentivirus, but in the 2000s when this was a new therapy, it was a modified HIV. Things get safer over time!
Just to make clear this treatment will always be in the "you'll survive next week? This treatment is too risky" class.