The main issue is that our immune system is very good at removing non-self, so a lot of work would need to go into stopping the immune system doing this.
The article isn't talking about CAR-T though, rather TCR-modified T cells. These are very patient-specific because we have a unique MHC signature (this is why we have to find a 'match' for a transplant recipient). Previously people have taken the T cells from the tumour site, boosted them in the lab, and re-infused with some remarkable and curative results. The outcomes in this trial weren't so great but it is novel to splice in a new TCR.
They are then further modified to go looking for specific targets (they go through your body, randomly, and check for specific molecules on cell membranes), and become "cytotoxic" towards them (they fire exploding acidic "bubbles" that digest a target cell).
Needless to say, this is a very dangerous treatment (uncontrolled killer T-cells roaming around inside your body multiplying ... not hard to imagine what happens if they target normal cells) and small mistakes will kill patients. In addition to directly killing off important cells, they may can also make the immune system overactive (which in fact happened in this study to three of the patients). They have a bad reputation in research for killing off entire batches of test animals when making a small mistake in less time than you'll need to diagnose the problem and, uh, "fix" it (you're supposed to kill test animals when a treatment doesn't work so they don't suffer). This is not allowed, and has to be explained to the authorities when it does happen and has ended research careers.
Normally, or should I say ideally, after multiplying a more-or-less set number of times, they die off and are removed by the body.
So no, this does not modify the bone marrow of the patient. Not at all. It just attacks and digests specific cells inside your body and dies off.
Just to make clear this treatment will always be in the "you'll survive next week? This treatment is too risky" class.
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!
(If you made it up I will be forced to feed it into the OpenAI playground to read the rest of the story...)
An oppressive government makes a copy of your consciousness, tortures it. Learns everything. You never know it happened.