Armitage-Doll was a nice advance over previous models, but it's also incomplete (and probably flat wrong in some cases, although working on pediatric malignancies has convinced me that a second cooperating event usually is mandatory).

In normal stem cells, it appears that attrition and immune clearance gets rid of damaged cells when they cycle and senescent cells all the time (subject to some variation, not entirely age related, at least in our volunteers). We may expect higher rates in filter organs, but liver cancer isn't too common, and the paper I referenced earlier shows that this can't be just an issue of fewer divisions (I despise the oversimplified Tomasetti & Vogelstein paper because the facts simply don't support it). Colorectal is probably more common because the crypts are "facing out" ala melanocytes, thus prone to accumulating lots of environmental damage.

Anyways, the latter of your possibilities (proliferative mutants divide faster and error more often than normal counterparts) makes the most sense -- the eventual "winner" in a tumor is the cell that produces the most progeny and resists apoptosis due to stress the best. It's probably not a coincidence that these are traits which adapt a mutated cell to survive chemotherapy as well. However, spawning nonself mutations willy-nilly is a great way to attract immune attention -- particularly if you haven't blown the immune system away by nuking it with chemotherapy. :-/