Also of note: just like good ol' mAbs, you need a target for CAR-T cells. And BiTEs, and BiKEs, and what have you.
Also also of note: the approaches that took pediatric ALL and adult APL (among others) from 5-10% survival to 90-95% survival weren't single pronged. (Good thing, too; I've seen patients who seemed ideal for CAR-T die when their malignancy outran the engineered cells) The majority of progress in real-world outcomes has arisen from combination or sequential therapies, incrementally improved to decrease toxicity and maximize efficacy.
Which is to say, incrementally advanced.
Last: per the above, you can use the two together. If a tumor is actively suppressing T cells, it isn't going to dissolve from CAR-T cells without a fight. There are additional tactics useful for "priming" the immune system, but in the end I don't see a lot of solid tumors melting away in trials of CAR-T cells (feel free to correct me).
Do take a look at the Kite trials and how they're structured. When you have to condition a patient with fludarabine to hit outcomes (at all) that's not a wonder drug. It's another tool in the toolbox. A necessary and useful tool, but a tool, not a magic wand.