Genetic custom nanites with hard-wired DNA code:
So part of the problem with DNA repair is that if it happens too fast, the cells start to engage more radical solutions to fix damage - sloppier repairs, because alive and mutated is better than dead. This is fine for a bacteria - either the repair works, and it lives, or it fails and dies. For a multicellular organism, the cell that lives but grows out of control is cancer. Any actual DNA can be altered, so let's overwrite DNA.
Your organelle can hold a copy of the person's DNA code, but made out of some coding material that is essentially indestructible at a cellular level and unaffected by radiation and mutation. Everyone gets a gene scan as a child, and receive injections of manufactured nanites programmed with their own code as needed. Got cancer? Get injected, and the nanites enter the cells and rewrite the DNA to the original code. Cancer stops. You're going into space and there's lots of radiation? Inject in advance, and add nanites as new cells grow. The nanites could also be tied to a tiny machine implanted that can make new nanites, but my thought is you want ZERO chance of mistaken code, so external and unalterable is better.
The nanites avoid the need for apoptosis, although a targeted form could still be used to rebuild damage from tumor growth (digest tumor, lay down new extracellular matrix for stem cells). The body has lots of mechanisms to slow growth and replication that have downsides in adults as growth slows. The people with these could heal faster safely, possibly even regrow lost tissue. Imagine being resilient like a small child all your life. This could even immortalize people regularly treated and reverse aging. Imagine a corporation selling immortality, and you always need another shot to counteract cells that replicate quickly. Cha-ching.
A slightly scary alternative would be nanites without code, that were connected to an external input with the patient's code. Wirelessly, the patient receives coding allowing a doctor to repair damage and even alter the person's DNA to respond to changing conditions. An unscrupulous corporation or government could insert DNA for obedience, alter their employees to live in alien environments and then leave them there (unable to return because they're no longer Earth-compatible), etc. The evil possibilities are endless.