That's easy to fix. If a cell has not just the existing error correction codes but also digital ones as well, then mutagenic substances (of which there are a lot) and telemere shortening can be fixed. Well, once we've figured out how to modify the DNA in-situ. Nanotech should have that sorted soonish.
The existing error correction is neither very good nor very reliable. This is a good thing, because it allows evolution. You don't want good error correction between generations. You just want it in a single person over their lifespan, and you want it restricted so that it doesn't clash with retrotranspons and other similar mechanisms. So, basically, one whole inter-gene gap/one whole gene protected by one code. Doable. You still need cell death - intercept the signal and use a guaranteed method.