It's not just a matter of brakes, it's more complicated than that. What happens is that a cancer cell doesn't just lose those control mechanisms - it also gains new and very aggressive abilities. It's like someone removed the brakes from a car AND then installed a much more powerful engine. The cell starts reproducing constantly because its growth instructions get stuck in the "on" position, completely ignoring the signals that would normally say "stop here". On top of that, they can live much longer than they should - normal cells have a limit to how many times they can divide, but cancer cells get around that.
There's a practical detail worth knowing if your grandfather is undergoing treatment: during chemotherapy or radiotherapy, precisely because these cells divide so fast they're easier to kill with those treatments. That's a positive side - their rush is also a weakness we can exploit. But that means the whole body suffers because we're trying to catch cells that move around a lot, so the side effects are often worse than if we were dealing with slower cells.
What really makes all this even more complicated is that cancer cells can also "hide" from the immune system - they produce substances that trick the body, cause new blood vessels to form and feed them, and can even deal better with lack of oxygen and nutrients. It's a combination of several bad superpowers at the same time, not just a single problem.