No, LNT is not accurate. It's accurate at high levels of exposure, not not low. In fact, there is growing evidence that low levels of exposure actually have health benefits (this is not saying to go get irradiated, as we don't have enough data).
LNT works under the assumption there is no biological repair mechanism. As you say, we are in a shooting gallery of radiation exposure. If we did have a way to handle low levels of radiation exposure then we wouldn't be alive. LNT causes more harm than it does good, because it causes over-reactions. It's the same reason breast cancer screening isn't recommended below a certain age. At a certain point, prevention does more damage than it helps because what you're preventing is so infrequent that the checks are more harmful than the chance you actually prevent something.
It really isn't a good match. It makes no sense. It does not assume there's a repair mechanism, as it's cumulative over a lifetime. If you include a repair mechanism then exposure rate needs to be included, not all time total exposure. We have models that take this into account and are more accurate at pridicting cancer risk.
I don't know if you even know what you're talking about if you don't know this. Even the Wikipedia page for linear no threshold tells you this in the first paragraph.